Optimal balking strategies and pricing for the ... - Semantic Scholar
Queueing Syst (2008) 59: 237–269 DOI 10.1007/s11134-008-9083-8
Optimal balking strategies and pricing for the single server Markovian queue with compartmented waiting space Antonis Economou · Spyridoula Kanta
Received: 24 April 2007 / Revised: 29 July 2008 / Published online: 3 September 2008 © Springer Science+Business Media, LLC 2008
Abstract We consider the single server Markovian queue and we assume that arriving customers decide whether to enter the system or balk based on a natural rewardcost structure, which incorporates their desire for service as well as their unwillingness to wait. We suppose that the waiting space of the system is partitioned in compartments of fixed capacity for a customers. Before making his decision, a customer may or may not know the compartment in which he will enter and/or the position within the compartment in which he will enter. Thus, denoting by n the number of customers found by an arriving customer, he may or may not know n/a + 1 and/or (n mod a) + 1. We examine customers’ behavior under the various levels of information regarding the system state and we identify equilibrium threshold strategies. We also study the corresponding social and profit maximization problems. Keywords Queueing · Balking · Equilibrium strategies · Social optimization · Profit maximization · Nash equilibrium · Pricing · Compartmented waiting space · Partial information Mathematics Subject Classification (2000) 60K25 · 90B22
A. Economou () · S. Kanta Department of Mathematics, Section of Statistics and Operations Research, University of Athens, Panepistemioupolis, Athens 15784, Greece e-mail: [email protected] S. Kanta e-mail: [email protected]
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1 Introduction In the queueing literature there exists an emerging tendency to study systems from an economic viewpoint. More concretely, after studying some performance measures of a system, a certain reward-cost structure is imposed and the objective is the optimization of the system. This control is generally classified as nonadaptive (static) or adaptive (dynamic). In nonadaptive control the objective is to choose the parameters of the system once and for all (e.g. number of servers, servers’ speed, waiting capacity etc.) so that the system behaves optimally. The appropriate mathematical tool for nonadaptive control is the classical optimization theory (non-linear programming). In adaptive control the objective is to choose an optimal policy for the administrator of the system for taking decisions as he observes the evolution of the system (e.g. to accept or not customers, to activate or not a secondary server etc.). In this case, the appropriate mathematical tools lie in the area of Markov decision processes (dynamic programming). Note, however, that in these two types of control, the administrator of the system solves the optimization problem and the customers are forced to follow his solution. If we drop the idea of central planning and we assume that there exists a rewardcost structure for the customers who are allowed to take their own decisions (e.g. to join or balk, to buy priority or not etc.) then the optimization problem can be viewed as a game among the customers. Then the first problem is to find equilibrium strategies for the customers. In a second level, the administrator has to solve the social and the profit maximization problems, taking into account the customers’ behavior. In this case of control, the customers’ behavior is analyzed using a game-theoretic framework and the administrator’s problems are solved by applying classical optimization techniques. In some problems the whole situation should be analyzed as a game between the customers and the administrator. These ideas go back at least to the pioneering works of Naor [24] and Edelson and Hildebrand [10] who studied individual, social and profit maximizing strategies for whether to join or balk in an M/M/1 queue with a simple reward-cost structure. Naor [24] studied the observable case in which each customer knows the number of customers in front of him before his decision, while Edelson and Hildebrand [10] studied the unobservable case. The corresponding results were refined by a number of authors, see e.g. [7, 8, 13, 21, 26]. Over the last years, research on equilibrium, social and profit maximizing strategies in queueing systems has grown considerably. Several authors have investigated such problems in queueing systems incorporating diverse characteristics as priorities, reneging, jockeying, schedules, retrials etc. (see e.g. [3, 9, 12, 17, 19, 23]). Hassin and Haviv [18] summarize the fundamental models and results in this area with extensive bibliographic references. Of particular interest is the value of information available to the customers at their decision epochs. Several papers compare the effect of the information level in the customer strategies and the performance of the system, see e.g. [1, 6, 8, 11, 13, 14, 16, 22]. Recently, the effect of the information level on the balking behavior of the customers has been extensively studied in the framework of call centers, see e.g. [4, 5, 20]. For a recent review of call center theory with many references in the role of information on the balking behavior of the customers see [2], Sect. 3 (demand modulation).
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The study of the balking behavior in the single server Markovian queue (M/M/1 queue) under various levels of information still remains a central problem. Recently, Hassin [15] studied the effect of information and uncertainty on profits in an unobservable M/M/1 queue. He considered scenarios in which the service rate, the service quality, or the waiting conditions are random variables that are known to the server but not to the customers and he asked whether the server is motivated to reveal these parameters. In another direction, Guo and Zipkin [11] studied the M/M/1 queue with a general cost structure under three levels of delay information: no information, partial information (the system occupancy) and full information (the exact waiting time). They identified cases where more accurate delay information improves performance and other cases where a more accurate information implies negative effects for the service provider or the customers. In the present paper we study the balking behavior of the customers in an M/M/1 queue with compartmented waiting space. Again we have a kind of partial information since the customers do not know the exact number of customers that they find upon arrival. Our aim is to complement the existing literature on the effect of information to the balking behavior of the customers by analyzing the effect of the compartmentalization. We present practical motivation for the study of the compartmentalization effect just after the detailed description of the model in Sect. 2. More specifically, we consider the M/M/1 queue and we suppose that the space of the system is partitioned in compartments of fixed capacity for a customers. Each arriving customer decides whether to join in the system or balk. Before making his decision, he may or may not know the compartment in which he will enter and/or the position within the compartment in which he will enter. Let n be the number of customers found by an arriving customer. Then the knowledge of the compartment in which he will enter corresponds mathematically to the knowledge of n/a + 1, while the knowledge of the position within the compartment in which he will enter corresponds to (n mod a)+1. In case where a customer knows both, he knows exactly n and the problem reduces to the fundamental observable model of Naor [24]. If he doesn’t know either, then the problem reduces to the fundamental unobservable model of Edelson and Hildebrand [10]. In this paper we consider separately the cases where all arriving customers observe only n/a + 1 (known compartment number—N case) or only (n mod a) + 1 (known compartment position—P case). We find the equilibrium strategies that are of threshold type. But then the administrator can impose an entrance fee and induce threshold type strategies that maximize the social benefit or his profit. We first show how the various optimal thresholds can be computed efficiently and we provide several qualitative results. We then compare the effect of the information level and of the various parameters of the system on the behavior of the customers and the performance of the system. The paper is organized as follows. In Sect. 2, we describe the parameters and the dynamics of the model, the reward-cost structure and the decision assumptions of the customers. In Sect. 3 we analyze the N case. More specifically we determine equilibrium, social and profit maximizing threshold strategies and we prove a certain ordering for the corresponding thresholds. In Sect. 4 we discuss the qualitative implications of the results and we also present several numerical scenarios. In Sect. 5 we
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carry out the analysis for the P case, while in Sect. 6 we comment on it, by taking a glance to several numerical studies. Finally, in Sect. 7 we summarize the conclusions regarding the influence of the customers’ information on the behavior of the system and we compare the various information cases.
2 The model Customers arrive at a service facility one by one according to a Poisson process with rate λ. There is one server, infinite waiting room and the service times of the customers are assumed to be independent identically distributed exponential random variables with rate μ, independent of the arrival process. We denote by ρ = μλ the traffic intensity of the system. The space of the system is partitioned in compartments of fixed capacity for a customers. The customers are processed on a FCFS basis. The state of the system is described by the process {N (t)} giving the number of customers in system. We are interested in the behavior of the customers when they can decide whether to join or balk upon their arrival. We suppose that there exists a reward of R units for every customer that completes a service in the system. This reward quantifies his satisfaction and/or the added value of being served. There exists also a waiting cost of C units per time unit as long as a customer remains in the system (in queue or in service). This cost quantifies his unwillingness to wait and/or lost benefits. The customers are assumed risk-neutral, i.e. their objective is to maximize the expected value of their net benefit without bothering for its variance. Under the above framework we can think of the situation as a symmetric game among the customers since they are all indistinguishable. Denote the common set of strategies and the payoff function by S and F , respectively. More concretely let F (a, b) be the payoff of a customer that selects strategy a when everyone else selects strategy b. A strategy se is a (symmetric Nash) equilibrium if F (se , se ) ≥ F (s, se ), for every s ∈ S. This means that it is a best response against itself, i.e., if all customers agree to follow it no one can benefit by changing it. A strategy s1 is said to weakly dominate strategy s2 if F (s1 , s) ≥ F (s2 , s), for every s ∈ S and for at least one s the inequality is strict. A strategy s∗ is said to be weakly dominant if it weakly dominates all other strategies in S. In the next two sections we obtain equilibrium customer strategies for joining/balking when the customers have partial information about the number of customers in the system. More concretely, the state-space of the system is written as {0, 1, 2, . . .} = ∞ i=1 Ki , where Ki = {n : (i − 1)a ≤ n < ia} and we consider separately two information cases for the customers: N: Known Compartment Number. Customers observe the number of the compartment in which they are going to enter but not the position within it. More specifically, if there exist n customers in the system just before the arrival of a tagged customer, his information will be the compartment number i = n/a + 1 in which he enters if he decides to join the system.
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P: Known Compartment Position. Customers observe the position of the compartment in which they are going to enter but they are not informed about the number of the compartment. The information of an arriving customer is the position i = (n mod a) + 1 in which he enters if he decides to join the system. We finally assume that the decisions of the customers are irrevocable, i.e. retrials of balking customers and reneging of entering customers are not allowed. The compartmentalization of the waiting space described above may reflect the way that the arriving customers observe the system or its real internal structure. As an example of the former case, consider an M/M/1 service system where the customers are forced to wait in a zigzag queue (like the ones in the airports, ticket offices etc.) In such cases the queue folds several times and every fold can be considered as a compartment of approximately a customers. An arriving customer may observe the number of folds in front of him so we have the N case. Alternatively, he may observe just the number of the customers in the last fold of the queue so we have the P case. As an example of the latter case, where the compartmentalization is related to a certain internal structure of the service space, let us consider a system which consists of a different queues and a unique server which operates as follows: The arrivals are assigned in the queues cyclically in a FCFS basis, i.e., the first customer in queue 1, the second customer in queue 2, . . . , the ath customer in queue a, the (a + 1)th customer in queue 1 and so on. The server provides service in a similar way. He serves sequentially the first customer of the queues 1, 2, . . . , a and after the service completion of the first customer in queue a, returns to the first one in queue 1 and so on. This mode of operation preserves the FCFS service discipline of the system and it seems convenient for some service systems where the customers can move very slowly, while the server is very flexible. For example think of a repair facility, where the arriving customers (cars, machines to repair etc.) are assigned to a queues (service places) and the server (repairman) considers the queues cyclically, serving only the first customer of every queue and then proceeding to the next queue. This mode of operation enables the server to use his time more efficiently because he does not have to wait for the upload of the next machine but rather goes to the next queue etc. When he returns in the queue the upload of the next machine to be served has been completed and he can start the service process immediately. In such a system an arriving customer may be informed about the number of items in the queue in which he is assigned. This corresponds to the N case of the model. On the other hand he may be informed just about which queue he will be assigned to and the current position (queue) where the server works. This corresponds to the P case of the model.
3 The known compartment number case: the results Suppose that we are in the framework of the known compartment number (N) case. Then a pure strategy is specified by a set A ⊆ {1, 2, . . .} which shows the ‘favorable’ compartment numbers for a customer, i.e. a customer decides to join the system if he knows that he will enter to a compartment with a number belonging to A. First, we are interested in conditions on the parameters of the model that ensure that the system
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is non-degenerate, i.e. some customers are willing to enter and the system is not continuously empty. Second, we seek for equilibrium pure strategies. We summarize the relevant results in the following. Theorem 3.1 The compartmented M/M/1 queue under the N information is nondegenerate if and only if 1 1 Rμ + ≥ 1, − aC 1 − ρ a a(1 − ρ) Rμ a − 1 + ≥ 1, aC 2a
if ρ = 1,
if ρ = 1.
(3.1) (3.2)
There exists a unique dominant pure strategy (which is also equilibrium). It is the threshold strategy ‘Enter if the compartment number that you will be assigned is less N = x N ’, with than or equal to iind ind Rμ N xind
=
aC Rμ aC
+
1 1−ρ a
+
a−1 2a
−
1 a(1−ρ)
if ρ = 1, if ρ = 1.
(3.3)
Proof Consider a tagged customer and suppose that all other customers follow a strategy A. Then the Markov chain describing the number of customers in the system will be eventually absorbed in the set {0, 1, 2, . . . , i ∗ a}, where i ∗ is the maximum integer such that {0, 1, 2, . . . , i ∗ } ⊆ A. Indeed, a moment of reflection shows that under strategy A all other states become transient and the system behaves as an M/M/1/i ∗ a queue. As it is well known, the stationary distribution of this queue is given by ⎧ n ⎨ (1−ρ)ρ , n = 0, 1, . . . , i ∗ a, if ρ = 1 ∗ 1−ρ i a+1 (3.4) P [Q = n] = ⎩ 1 , n = 0, 1, . . . , i ∗ a, if ρ = 1. ∗ i a+1 Consider now the tagged customer who is to decide whether to join the system or not, given the information that he can enter in the compartment number i. So he knows that the number of customers at his arrival is n ∈ {(i − 1)a, (i − 1)a + 1, . . . , ia − 1}. If he decides to enter then his expected net individual benefit is N Sind (i) = R −
C {E[Q− |Q− ∈ {(i − 1)a, (i − 1)a + 1, . . . , ia − 1}] + 1}, μ
(3.5)
where Q− is a random variable having the equilibrium distribution of the number of customers at an arrival instant in the M/M/1/i ∗ a queue. Because of the PASTA property we have that the distribution of Q− coincides with that of Q in (3.4). Using (3.4) in (3.5) we obtain that N Sind (i) =
R− R−
C μ {ia C μ {ia
− −
a 1 1−ρ a + 1−ρ }, a−1 2 },
if ρ = 1, if ρ = 1.
(3.6)
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N (i) ≥ 0, that is when The tagged customer will decide to enter if and only if Sind Rμ 1 1 a−1 i ≤ Rμ aC + 1−ρ a − a(1−ρ) , for ρ = 1 or when i ≤ aC + 2a , for ρ = 1. Hence, the N = x N with x N given by (3.3) is a dominant threshold strategy with threshold iind ind ind strategy. To ensure that the system does not remain for ever empty we must have N = 0 which gives the conditions (3.1) and (3.2). iind
We now consider the problem of social and profit maximization. By imposing an admission fee, the administrator of the system can force the customers to adopt any threshold i he may desire. Indeed, for a desired i he must set the admission fee p such that (R−p)μ 1 1 aC + 1−ρ if ρ = 1, a − a(1−ρ) i= (3.7) (R−p)μ aC + a−1 if ρ = 1. 2a N that maximizes the mean social net profit per time unit. We seek for a threshold isoc This is given in the next theorem.
Theorem 3.2 In the compartmented M/M/1 queue under the N information, the threshold strategy that maximizes the social net profit per time unit is ‘Enter, if the N ’, where compartment number that you will be assigned is less than or equal to isoc N N = x N , x N being the unique solution of the equation g(x) = x isoc soc soc ind in [1, ∞] with ⎧ xa+1 ) 1 1 ⎨ (xa+1)(1−ρ a )−a(1−ρ + 1−ρ if ρ = 1 a − a(1−ρ) a(1−ρ)(1−ρ a ) g(x) = (3.8) ⎩ a x 2 − a−2 x if ρ = 1. 2
2
Moreover the optimal social threshold is smaller than or equal to the optimal individual threshold, i.e. N N ≤ iind . isoc
(3.9)
N (i) = λ∗ R − Proof For any threshold i, the social net profit per time unit is Ssoc ∗ CE[Q], where λ is the mean arrival rate and E[Q] the mean number of customers in system, given that the customers follow the threshold policy with threshold i. In this case the system behaves as an M/M/1/ia queue and using the stationary distribution given by (3.4) we obtain
N (i) = Ssoc
⎧ ⎨ λR
1−ρ ia ρ − C{ 1−ρ 1−ρ ia+1 ⎩ λR ia − C ia , ia+1 2
−
(ia+1)ρ ia+1 }, 1−ρ ia+1
if ρ = 1, if ρ = 1.
(3.10)
N (i) is unimodal with Suppose now that ρ = 1. We will prove that the function Ssoc N respect to i and that isoc as defined in the statement of the theorem is a local maximum N (i); hence a global maximum. To this end we study the increments S N (i) − of Ssoc soc
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N (i − 1). After some long but tedious calculations we obtain that Ssoc N N (i) − Ssoc (i − 1) ≥ 0 Ssoc
⇔
(ia + 1)(1 − ρ a ) − a(1 − ρ ia+1 ) Rμ ≤ a(1 − ρ)(1 − ρ a ) aC
⇔
N , g(i) ≤ xind
(3.11)
N are given by the first branches of (3.8) and (3.3) respectively. To where g(i) and xind N (i), it is now sufficient to prove that g(x) is increasing. prove the unimodality of Ssoc To this end, define the function f : [1, ∞) → R with f (x) = g(x) − x. We have ρ ρ f (x) = 1−ρ + (1−ρaaln)(1−ρ) ρ xa+1 and we can easily see that f (x) ≥ 0 for x ≥ 1. Hence we have that g(x) is increasing in [1, ∞) and limx→∞ g(x) = ∞. Note also N (i) is unimodal. In light of (3.11), its global maximum i N is that g(1) = 1. So Ssoc soc characterized by N N N ) ≤ xind ≤ g(isoc + 1). g(isoc
(3.12)
N = x N , where x N is the unique solution of the equation Hence we have that isoc soc soc N . g(x) = xind It remains now to prove (3.9). Because f (x) is increasing in [1, ∞) and f (1) = N ) ≥ 0, i.e. g(x N ) − x N = x N − x N ≥ g(1) − 1 = 0 we have in particular that f (xsoc soc soc soc ind N ≤ iN . 0. But then taking floors we have isoc ind The case where ρ = 1 is done similarly using the second branches of (3.3), (3.8) and (3.10).
We now proceed to the profit maximization problem. Here, the administrator of the system imposes an entrance fee for maximizing his own benefit per time unit. In the following theorem we provide the profit maximization threshold. Theorem 3.3 In the compartmented M/M/1 queue under the N information, the threshold strategy that maximizes the administrator’s net profit per time unit is ‘Enter, N ’, if the compartment number that you will be assigned is less than or equal to iprof N = x N , x N being the unique solution of the equation h(x) = x N in where iprof prof prof ind [1, ∞] with xa−a )(1−ρ xa+1 ) x + (1−ρ , if ρ = 1, ρ xa−a (1−ρ)(1−ρ a ) (3.13) h(x) = x + (x − 1)(xa + 1), if ρ = 1. Moreover the optimal profit maximization threshold is smaller than or equal to the optimal social threshold, i.e. N N iprof ≤ isoc .
(3.14)
Proof In light of (3.7) the maximum entrance fee that can be imposed by the administrator of the system in order to force the customers to adopt a given threshold i is ⎧ 1 1 ⎨ R − aC μ (i − 1−ρ a + a(1−ρ) ), if ρ = 1, p(i) = (3.15) ⎩ R − aC (i − a−1 ), if ρ = 1. μ 2a
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245
N (i) = λ∗ p(i) where λ∗ is the mean Then his expected profit per time unit is Sprof arrival rate given that the customers follow the threshold i. The system behaves as an M/M/1/ia queue and therefore using (3.4) we obtain after some algebra that
N Sprof (i) =
⎧ ia ) N ⎨ λaC(1−ρ ia+1 (xind − i), μ(1−ρ
⎩
)
λa 2 Ci N μ(ia+1) (xind
− i),
if ρ = 1, (3.16) if ρ = 1,
N is given by (3.3). where xind Suppose now that ρ = 1. Then, after some calculations we obtain that N (i) Sprof N (i − 1) Sprof
≥1
⇔
N h(i) ≤ xind ,
(3.17)
N are given by the first branches of (3.13) and (3.3) respecwhere h(i) and xind tively. The function h(x) is easily seen to be increasing in [1, ∞), h(1) = 1 and N (i) is unimodal and its maximum i N is characterized by limx→∞ h(x) = ∞ so Sprof prof N N N h(iprof ) ≤ xind ≤ h(iprof + 1).
(3.18)
N = The function h(x) is continuous so we have that the optimal threshold is iprof N , where x N is the unique solution of h(x) = x N in [1, ∞). xprof prof ind N (i) < 0 We now proceed to prove (3.14). From (3.16) we have obviously that Sprof N so we have immediately i N ≤ i N . We have also established that i N ≤ for i > iind soc prof ind N in (3.9). Therefore in the case where i N = i N we have immediately (3.14). iind soc ind N < i N . It suffices to show that for integers i with i N ≤ i < i N , Suppose that isoc soc ind ind N N (i), i.e. S N (i) is decreasing in {i N , i N + 1, . . . , i N }. we have Sprof (i + 1) ≤ Sprof soc soc prof ind N (i), we will have then (3.14). Because of the unimodality of Sprof N , we have from the monotonicity of g(x) given in (3.8) and from (3.12) For i ≥ isoc N . To prove that S N (i + 1) ≤ S N (i) we have to prove in light of that g(i + 1) ≥ xind prof prof N . So it suffices to prove that h(i + 1) ≥ g(i + 1), i.e. (3.17) that h(i + 1) ≥ xind
i
1 ρ a (1 − ρ ia+1 ) 1 (1 − ρ ia+1 )(1 − ρ ia+a ) − + . ≤ i + 1 − ρ (1 − ρ)(1 − ρ a ) 1 − ρ a ρ ia (1 − ρ)(1 − ρ a )
(3.19)
This reduces after some simplification to prove i
1 − ρ ia ρ ≤ ia . 1 − ρ ρ (1 − ρ)(1 − ρ a )
(3.20)
For ρ < 1, (3.20) is written equivalently as i≤
1 1 1 1 − ρ ia = + + · · · + ia+1 , ρ ia+1 (1 − ρ a ) ρ a+1 ρ 2a+1 ρ
(3.21)
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which is clearly valid. For ρ > 1, (3.20) is written similarly as i≥
1 1 − ρ ia 1 1 = + + · · · + ia+1 , ρ ia+1 (1 − ρ a ) ρ a+1 ρ 2a+1 ρ
(3.22)
which is again valid. The case where ρ = 1 is treated similarly using the second branches of (3.3), (3.13) and (3.16). By solving (3.3), (3.12) and (3.18) with respect to R and using the results of Theorems 3.1, 3.2 and 3.3 we have the following figures that show the equilibrium, social optimal and profit maximizing thresholds under the N information as R varies. N versus R. (a) Equilibrium pure strategies: Thresholds iind
1
N (1) tind
2
• N (2) tind
3
•
•
•
•
•
N (3) tind
where N tind (i) =
N iind
R
⎧ ⎨C μ {ai −
a 1−ρ a
⎩ aC (i −
a−1 2a ),
μ
+
1 1−ρ },
ρ = 1, ρ = 1.
N versus R. (b) Social Optimization: Thresholds isoc
1
N (1) tsoc
2
• N (2) tsoc
•
3
•
•
•
•
N (3) tsoc
where N (i) = tsoc
N isoc
R
⎧ a ia+1 ) ⎨ C (ia+1)(1−ρ )−a(1−ρ , a
ρ = 1,
⎩ aC {ai 2 − (a − 2)i −
ρ = 1.
μ
(1−ρ)(1−ρ )
2μ
a−1 a },
N versus R. (c) Profit Maximization: Thresholds iprof
1
N (1) tprof
• N (2) tprof
2
• N (3) tprof
3
N iprof
•
•
•
• R
Queueing Syst (2008) 59: 237–269
where
⎧ ⎨ aC {i +
μ N tprof (i) = ⎩ aC {i μ
247
(1−ρ (i−1)a )(1−ρ ia+1 ) ρ (i−1)a (1−ρ)(1−ρ a )
+ (i − 1)(ia + 1) −
−
1 1−ρ a
a−1 2a },
+
1 a(1−ρ) },
ρ = 1, ρ = 1.
4 The known compartment number case: discussion In this section we discuss some qualitative properties and implications of the results that concern the N case. Moreover, we present some numerical results to illustrate the dependence of the optimal thresholds on the parameters of the model. 4.1 The effect of the externalities The first remark concerns the discrepancy between individual and social optimization in the N case. Indeed the relation (3.9) shows that individual optimization leads to longer queues than are socially desired. This happens because an arriving customer that decides to enter the system ignores the negative externalities that imposes on future arrivals. Indeed such a customer may join the queue even if his own expected benefit is smaller that the expected reduction in the benefit of future customers. The same situation occurs in [24] model. Hassin [13] described an equivalent model operating under the LCFS-PR discipline, where the customers do not impose externalities and they behave socially optimally. This result shows that the externalities constitute the unique source for the discrepancy between the socially and the individually optimal behavior of the customers. We will now show that the same reasoning can be extended in the framework of our model. More specifically we show that in an equivalent model with the LCFS-PR discipline the customers do not impose externalities on the others and their individual optimal threshold coincides with the social optimal threshold given in Theorem 3.2. More concretely, we suppose that newly arriving customers join the system and start immediately being served, possibly preempting the service of another customer. Moreover, when a preempted customer reenters for service, his service is continued from the point of interruption. Regarding the information, we suppose that the customers know continuously the compartment in which they reside but not the position within it (N case). The relevant decision that an individual faces is when to leave the system. Because of the memoryless property of the exponential distribution and the fact that a customer knows continuously only the compartment in which he resides, a moment of reflection shows that the customer has incentive to leave the system only at epochs when he is moved to another compartment because of a new arrival. If the customers follow a threshold strategy i ∗ , i.e. they do not abandon the system as long as they reside to a compartment less than or equal to i ∗ , then the system behaves as an M/M/1/i ∗ a queue. Moreover, because of the LCFS-PR discipline the customers do not impose externalities to other customers. Indeed, if we consider a tagged customer, it is clear he does not impose externalities to future customers, because all of them are placed in front of him in the queue. On the other hand, by the memoryless property of the exponential distribution and the homogeneity of the
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customers, we can see that the tagged customer may decide to renege, only when there are no customers behind him. For, if it is beneficial for the tagged customer to renege at some time, then a customer who is behind him at that time should have been removed earlier, when he was at the same position as the tagged customer now. Therefore, when a customer decides whether to renege or not, he imposes no externalities whatever he does, because everybody present is served prior to him as well as the future arrivals. We now determine the individually optimal policy for a customer under the LCFSN −LCFSPR PR discipline. Let Sind (i) be the expected net benefit of the last customer in the system (this is the only that can renege) given that he has just moved to the compartment i because of a new arrival and given that all customers renege when they enter the compartment i + 1 (including the customer under consideration). He then knows that he is exactly at the position (i − 1)a + 1 of the system (first in the compartment i). The customer prefers to stay in the system for all compartment N −LCFSPR numbers i such that Sind (i) ≥ 0. We have that N −LCFSPR Sind (i) = Rfi − CSi ,
(4.1)
where fi is the probability that a customer will reach state 0 (completion of service) before reaching state ia + 1 (abandonment), starting from the position (i − 1)a + 1 and Si is the corresponding mean remaining time till he leaves the system (either because of service completion or abandonment). The probability fi is the classical gambler-ruin probability, where the initial point is (i − 1)a + 1, the goal is ia + 1 λ and the winning probability in each round is p = λ+μ and q = 1 − p. Moreover, Si 1 is the expected number of rounds till the game is over multiplied by λ+μ (the mean time till the next transition). Using the well known formulas for these quantities (see e.g. [25], Sect. 4.4, Example 4.4(a)) we conclude that ⎧ a ⎨ 1−ρia+1 , if ρ = 1, 1−ρ fi = (4.2) ⎩ a , if ρ = 1, ia+1
and Si =
⎧ ⎨ ⎩
1 μ(1−ρ) {(ia a 2μ {(i
a
1−ρ + 1) 1−ρ ia+1 − a},
− 1)a + 1},
if ρ = 1, if ρ = 1.
(4.3)
N −LCFSPR By plugging (4.2) and (4.3) in (4.1) and solving Sind (i) ≥ 0 for i we arrive after some straightforward calculations at (3.11). Hence we conclude that the individually optimal policy in this case coincides with the socially optimal policy of the initial model. Therefore, the externalities constitute the unique source for the discrepancy between the socially and the individually optimal behavior of the customers.
4.2 Dependence of the optimal threshold on the model parameters In this section we present some numerical results to illustrate the behavior of the optimal thresholds iind , isoc , iprof for the N case when keeping all parameters fixed and let only one vary. In all applications we consider the waiting cost C per time
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249
Fig. 1 Optimal thresholds with respect to R − N case
unit and the service rate μ to be equal to 1. This can be always assumed, without loss of generality, with appropriate re-scaling of the time and reward/cost units. So we first examine the behavior of the thresholds as R or λ varies. Then, when the compartment size a varies, we depict iind a, isoc a, iprof a (the maximum numbers of potential customers under the optimal policies) as functions of a. This is clearly more informative than depicting iind , isoc , iprof with respect to a. In Fig. 1 we present the equilibrium, social optimization and profit maximization thresholds with respect to R for λ = 0.7 and a = 4. As we can observe the three thresholds are all increasing ladder functions of R. The individual optimal threshold increases more rapidly than the other optimal thresholds. This is because of the form N (i) which are linear in i. The profit maximizing threshold increases very slowly of tind with respect to R. This fact can be explained due to the form of the critical numbers N (i) which increase exponentially fast. tprof Figure 2 shows the equilibrium, social optimization and profit maximization thresholds with respect to λ for R = 50 and a = 4. Note that the equilibrium threshold converges to Rμ aC as λ increases (because of (3.3)). The social optimization and profit maximization thresholds are unimodal functions of λ and they become ultimately 1 for large values of λ. In Fig. 3 we display the maximum number of potential customers for the corresponding equilibrium, social optimization and profit maximization thresholds with respect to a for λ = 0.7 and R = 50. As a increases above a certain level all thresholds become equal to 1. This justifies the linear part at the right part of the figure. In the meanwhile the thresholds are not monotone functions of a. For example in the curve of the equilibrium threshold we see that there exist many local maxima and minima. N a is near to the full information case equiIn particular we see that the minimum iind librium threshold (a = 1). The same value is also attained for a = 2, 5, 10, 25, 50, i.e.
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Fig. 2 Optimal thresholds with respect to λ − N case
Fig. 3 Maximum number of potential customers with respect to a − N case
for all values of a that are divisors of the full information case equilibrium threshold. This explains the non-monotone behavior of the optimal threshold with respect to a. The maximum number of potential customers in the system under the full information is 50. We expect that for other values of a the corresponding optimal threshold N a is also near 50. So when a = 20 we would expect the threshshould be such that iind
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251
Fig. 4 Optimal social benefit and administrator’s profit with respect to a − N case
old to be equal to 2 or 3. As we can observe the threshold is 3, letting 60 customers to enter the system. On the other hand for a = 25 the adequate threshold would drop to 2 so as to be in agreement with the full information case. This is exactly what happens. That difference between the two cases of a is due to the partial information that the customers receive in the N case. Same observations occur with the social and profit maximizing thresholds. The message for the designer of the system is that the compartmentalization should ‘agree’ with the optimal threshold if the customers had full information. The compartment size a should be chosen close to some divisor of the optimal threshold for the corresponding fully observable model. In the last Fig. 4, we present the social benefit and administrator’s profit, when the customers follow the corresponding optimal policies, with respect to a for λ = 0.9 and R = 25. For a = 1 which corresponds to full information of the customers about the state of the system, the optimal social benefit is high, while the administrator’s profit attains its minimum value. As we can observe for small values of a the difference of the two functions is positive, whereas for greater values of a the two functions coincide. Note that there is a value of a, in our example this value is a = 7, such that the administrator’s profit is maximized and then it decreases. This is in some sense the ‘ideal’ compartment size for the administrator. Moreover, in that case the whole social benefit is taken by the administrator. Thus, we conclude that the administrator can improve its profit by an adequate selection of the compartment size. 4.3 Comparison with Naor’s observable model A possible interpretation for the N case is that the customers take their decisions whether to join or to balk, having only a rough information about the number of customers in the system. For example, for a = 12 or a = 100 the customers count
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how many dozens or hundreds of customers are in the system. So, it is reasonable for the form of the optimal policies in the N case to be similar to the form of the optimal policies in Naor’s [24] observable model as it is confirmed by Theorems 3.1, 3.2 and 3.3. On the other hand, the optimal thresholds in the N case are significantly different than the optimal thresholds of Naor’s [24] model. In Naor’s [24] model, the equilibrium threshold Rμ C does not depend on λ while the equilibrium threshold of the N case given in Theorem 3.1 clearly does. This is because the arrival rate does not offer any information to the customer in the observable model. However, in the N case, it influences customer’s estimate about his position in the queue (see (3.4) and (3.5)). The numerical studies for Naor’s [24] model regarding the effect of the arrival rate on the thresholds have been investigated by Hassin [13]. In the N case of our model we have that the optimal equilibrium threshold is decreasing in λ. For the social and profit maximizing thresholds the numerical results agree with the findings of Hassin [13] for Naor’s model, i.e. both thresholds are unimodal functions of the arrival rate. A noteworthy similarity with Naor’s [24] model is the existence of a dominant strategy in the N case. The existence of dominant strategies is typical only for fully observable models (see e.g. [6, 24]). In general, although equilibrium strategies do exist in partially observable models, dominant strategies do not exist. Indeed, in partially observable models, the customer’s net benefit function Sind (i) depends both on the state of the system and the strategy of the other customers. However, in the N case N (i) in (3.5) does not depend on the of the present model we have that the function Sind ∗ strategy i of the other customers. This happens because the conditional distribution of the queue length Q− given that Q− ∈ {(i − 1)a, (i − 1)a + 1, . . . , ia − 1} in an M/M/1/i ∗ a queue is the same for any i ∗ ≥ i. So the N case of the present model is a rare example where a partially observable model has a dominant strategy.
5 The known compartment position case: the results We now turn to the framework of the known compartment position (P) case. A pure strategy here is specified by a set A ⊆ {1, 2, . . . , a} which shows the favorable compartment positions for a customer, i.e. in this case a customer decides to join the system if and only if he will take a position belonging to A. Note here that the number of pure strategies is finite and equals 2a . We consider also mixed strategies. A mixed strategy is specified by a vector (q1 , q2 , . . . , qa ), where qi is the probability that a customer enters the system when he is given the information that he will be assigned to position i of the compartment. A mixed strategy with qi = 1 for 1 ≤ i ≤ j , qj +1 = q and qi = 0 for j + 1 < i ≤ a is referred to as a mixed threshold strategy with threshold j + q. In the special case where q = 0 we have a pure threshold strategy with threshold j . We establish conditions that ensure that the system is non-degenerate and we also determine the equilibrium mixed strategies. To this end we begin by studying the equilibrium behavior of the system given that all customers follow the same mixed strategy (q1 , q2 , . . . , qa ). Note that there are two cases that should be treated differently. The first case refers to strategies with qi = 0 for some i. In this case the Markov chain that describes the number of customers in the system is eventually
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253
Fig. 5 Transition diagram—P case (a)
Fig. 6 Transition diagram—P case (b)
absorbed to the finite set {0, 1, . . . , i ∗ }, where i ∗ + 1 is the first i with qi = 0, i.e. q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0. The corresponding transition diagram is given in Fig. 5. The second case covers strategies with qi > 0 for all i = 1, 2, . . . , a. Then the Markov chain is irreducible in the set N0 of the non-negative integers and the transition diagram is given in Fig. 6. Using the standard product-form formula for the stationary distribution of birthdeath processes we have the following. Lemma 5.1 Consider the compartmented M/M/1 queue under the P information. (a) If all customers follow a strategy (q1 , q2 , . . . , qa ) with q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0 for some i ∗ = 1, 2, . . . a − 1, then the queue is always stable and its stationary distribution is given by B, n = 0, P [Q = n] = (5.1) Bρ n q1 q2 . . . qn , n = 1, 2, . . . i ∗ , where
⎛
⎞−1
∗
B = ⎝1 +
i
ρ n q 1 q 2 . . . qn ⎠
.
(5.2)
n=1
(b) If all customers follow a strategy (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a then the queue is stable if and only if q1 q2 . . . qa ρ a < 1.
(5.3)
Then its stationary distribution is given by P [Q = n] = Bρ n (q1 q2 . . . qa )n/a
(n mod
a)
qj ,
n = 0, 1, . . . ,
(5.4)
j =1
where
a t −1 t=1 q1 q2 . . . qt ρ B = 1+ 1 − q 1 q 2 . . . qa ρ a
(where the product in (5.4) is interpreted as 1 for n ≡ 0 mod a).
(5.5)
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We are now in position to find all equilibrium strategies in the P case. We have the following. Theorem 5.2 The compartmented M/M/1 queue under the P information is nondegenerate if and only if R≥
C . μ
(5.6)
The best responses against any given strategy are always of threshold type (pure or mixed). Hence, the equilibrium strategies are of threshold type. More concretely, if ρ < 1, we have the following cases as R varies in [ C μ , ∞): (a) For R = i C μ for some i = 1, 2, . . . , a − 1 there exist multiple equilibrium strategies. They are the mixed threshold strategies with thresholds in [i − 1, i]. C (b) For R ∈ (i C μ , (i + 1) μ ) for some i = 1, 2, . . . , a − 1 there exists a unique equilibrium strategy. It is the pure threshold strategy with threshold i. C (c) For R ∈ [a C μ , a μ(1−ρ a ) ) there exists a unique equilibrium strategy. It is the threshold strategy with threshold a − 1 + (1 −
Ca −a Rμ )ρ .
C (d) For R ∈ [a μ(1−ρ a ) , ∞) there exists a unique equilibrium strategy. It is the pure threshold strategy with threshold a. If ρ ≥ 1, we still have the cases (a) and (b) as above, but instead of the cases (c) and (d) we have the case (c ) For R ∈ [a C μ , ∞) there exists a unique equilibrium strategy. It is the threshold
strategy with threshold a − 1 + (1 −
Ca −a Rμ )ρ .
Proof We are interested in finding the best responses for every given mixed strategy (q1 , q2 , . . . , qa ). To this end we consider a tagged customer and suppose that all other customers follow a strategy (q1 , q2 , . . . , qa ). We have to distinguish two cases because the stationary distribution of the number of customers has two different forms corresponding to the cases (a) and (b) of Lemma 5.1. We first consider the case where all customers follow a strategy (q1 , q2 , . . . , qa ) with q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0 for some i ∗ = 1, 2, . . . , a − 1. Now the system is always stable (see Lemma 5.1). We consider a tagged customer who is to decide whether to join or balk, given the information that he can enter in the position i of the compartment. So he knows that the number of customers at his arrival is exactly n = i − 1 and his expected net individual benefit is P (i) = R − Sind
C i. μ
(5.7)
P (i) is decreasing in i so we have that there exists an i such that The function Sind P P (i) < 0 for i = i + 1, . . . , a. Hence a best reSind (i) ≥ 0 for i = 0, 1, . . . , i and Sind P (i ) = 0 then every mixed threshold strategy
sponse is the threshold strategy i . If Sind P (i ) > 0 the with threshold i ∈ [i − 1, i ] is best response, while in the case with Sind
threshold strategy i is the unique best response against (q1 , q2 , . . . , qa ).
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255
We conclude that among strategies (q1 , q2 , . . . , qa ) with q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0 for some i ∗ = 1, 2, . . . a − 1, the only possible equilibrium is of the form (1, 1, . . . , 1, q, 0, 0, . . . , 0) with q ∈ (0, 1], i.e. mixed threshold strategies with thresholds in (0, a − 1]. Note now that a strategy (1, 1, . . . , 1, q, 0, 0, . . . , 0) with q ∈ (0, 1) P (i) = 0 which gives R = i C in the ith position can be equilibrium if and only if Sind μ for some i = 1, 2, . . . , a − 1. This corresponds to case (a) of the theorem. Similarly, a strategy (1, 1, . . . , 1, 0, 0, . . . , 0) with 1s up to the ith position is unique equilibrium P (i) > 0 and S P (i + 1) < 0. Solving for R we obtain case (b) of the if and only if Sind ind theorem. We now consider the other case of Lemma 5.1 and suppose that all customers follow a strategy (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a. If the system is unstable under this strategy, i.e. q1 q2 . . . qa ρ a ≥ 1 then the best response of the tagged customer is not to enter. Hence, in case that the condition (5.3) does not hold a strategy (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a cannot be an equilibrium. Suppose now that the condition (5.3) holds and the system is stable and consider a tagged customer who is to decide whether to join or balk, given the information that he can enter in the position i of the compartment. So he knows that the number of customers at his arrival is n ∈ {i − 1, a + i − 1, 2a + i − 1, . . .}. If he decides to enter then his expected net individual benefit is P (i) = R − Sind
C {E[Q− |Q− ∈ {i − 1, a + i − 1, 2a + i − 1, . . .}] + 1}, μ
(5.8)
where Q− is a random variable having the equilibrium distribution of the number of customers at an arrival instant. Because of PASTA property we have that Q− is distributed as Q in (5.4). Using (5.4) we compute the expected value in (5.8) and we obtain that the expected net individual benefit is C ρ a q 1 q 2 . . . qa P (i) = R − + i . (5.9) Sind a μ 1 − ρ a q1 q2 . . . qa P (i) is decreasing in i so we have that there exists an i such that The function Sind P (i) ≥ 0 for i = 0, 1, . . . , i and S P (i) < 0 for i = i + 1, . . . , a. Hence a best reSind ind P (i ) = 0 then every mixed threshold strategy sponse is the threshold strategy i . If Sind P (i ) > 0 the threshold strategy i
with threshold i ∈ [i − 1, i ] is best response. If Sind is the unique best response against (q1 , q2 , . . . , qa ). We conclude that among strategies (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a, the only possible equilibrium is of the form (1, 1, . . . , q) with q ∈ (0, 1], i.e. mixed threshold strategies with thresholds in (a − 1, a]. Note that under a policy (1, 1, . . . , q), (5.9) assumes the form ρa q C P a +i . (5.10) Sind (i) = R − μ 1 − ρa q P (a − 1) > 0 and S P (a) = 0 for the case To be an equilibrium we should have Sind ind P P (a − 1) ≥ 0 for R and then q ∈ (0, 1) or Sind (a) ≥ 0 for the case q = 1. Solving Sind C Ca P (a) = 0 for q gives R ∈ [a C , a −a which is the case Sind μ μ(1−ρ a ) ) and q = (1 − Rμ )ρ
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P (a) > 0 for R gives the case (d) of the theorem. Of (c) of the theorem. Solving Sind course for ρ ≥ 1 we should have q < ρ −a for stability and the case (d) does not occur. The case (c) becomes then (c ).
We can now proceed to the problem of social and profit maximization. Again by imposing an appropriate admission fee, the administrator of the system can oblige the customers to adopt any desired threshold i. So we seek threshold strategies that maximize the social net reward and the administrator’s profit per time unit. Theorem 5.3 In the compartmented M/M/1 queue under the P information define P (i), i = 1, 2, . . . , a by the numbers tsoc
P tsoc (i) =
⎧ C ⎪ ⎪ μ(1−ρ) {i − ⎪ ⎪ ⎪ ⎪ aC(1−ρ a ) ⎪ ⎨ μ(1−ρ) , i(i+1)C ⎪ ⎪ ⎪ 2μ , ⎪ ⎪ ⎪ ⎪ ⎩ a2 C μ ,
ρ(1−ρ i ) 1−ρ },
i = 1, 2, . . . , a − 1, ρ = 1, i = a, ρ = 1, i = 1, 2, . . . , a − 1, ρ = 1,
(5.11)
i = a, ρ = 1
and let qsoc defined by ⎧ a) ⎪ ⎨ ρ1a 1 − Ca(1−ρ Rμ(1−ρ) , qsoc = ⎪ ⎩ 1 − Ca 2 , Rμ
ρ = 1, (5.12) ρ = 1.
If ρ < 1, the threshold strategies that maximize the social net reward per time unit as R varies in [ C μ , ∞) are given as follows: P (i) for some i = 1, 2, . . . , a − 1 there exist multiple social optimal (a) For R = tsoc threshold strategies with thresholds in [i − 1, i]. P (i), t P (i + 1)) for some i = 1, 2, . . . , a − 1 there exists a unique so(b) For R ∈ (tsoc soc cial optimal threshold strategy. It is the pure threshold strategy with threshold i. aC P (a), (c) For R ∈ [tsoc μ(1−ρ)(1−ρ a ) ) there exists a unique social optimal threshold
strategy. It is the threshold strategy with threshold a − 1 + qsoc . aC (d) For R ∈ [ μ(1−ρ)(1−ρ a ) , ∞) there exists a unique social optimal threshold strategy. It is the pure threshold strategy with threshold a. If ρ ≥ 1, we still have the cases (a) and (b) as above, but instead of the cases (c) and (d) we have the case P (a), ∞) there exists a unique social optimal threshold strategy. It is (c ) For R ∈ [tsoc the threshold strategy with threshold a − 1 + qsoc . P (x) to be the social net profit per Proof For a given threshold x ∈ [0, a] define Ssoc ∗ time unit. Letting λ (x) and E[Q](x) be the mean arrival rate and the mean number of customers respectively, given that the customers follow the threshold policy x, we P (x) = λ∗ (x)R − CE[Q](x). Of course S P (0) = 0. have that Ssoc soc
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257
Suppose that ρ < 1. If x ∈ (0, a − 1] then x = i − 1 + q with i ∈ {1, 2, . . . , a − 1} and q ∈ (0, 1]. The stationary distribution of the system is given by (5.1) and so i−1 and E[Q](x) = we can compute after some algebra that λ∗ (x) = λ 1−(1−q(1−ρ))ρ 1−(1−q(1−ρ))ρ i ρ(1−ρ i )−i(1−q(1−ρ))ρ i (1−ρ) . (1−ρ)(1−(1−q(1−ρ))ρ i ) P1 Ssoc (x) =
P (x) assumes the form Then the function Ssoc
i−1 )] − (1 − q(1 − ρ))ρ i C[ Rμ − i] ρC[ Rμ C − (1 + ρ + · · · + ρ C , 1 − (1 − q(1 − ρ))ρ i
x = i − 1 + q, i ∈ {1, 2, . . . , a − 1}, q ∈ (0, 1].
(5.13)
Note that for x = 0, (5.13) is also valid for i = 1 and q = 0. If x ∈ (a − 1, a] then x = a − 1 + q with q ∈ (0, 1]. The stationary distribua−1 and tion of the system is given by (5.4) and we obtain λ∗ (x) = λ 1−(1−q(1−ρ))ρ 1−ρ a E[Q](x) =
aqρ a 1−qρ a
+
ρ 1−ρ
−
aρ a 1−ρ a .
P (x) assumes the form In this case the function Ssoc
1 − (1 − q(1 − ρ))ρ P2 (x) = λR Ssoc a
a−1
1−ρ
aqρ a ρ aρ a , −C + − 1 − qρ a 1 − ρ 1 − ρ a
x = a − 1 + q, q ∈ (0, 1].
(5.14)
Note that for x = a − 1 (5.14) is also valid for q = 0, that is (5.13) and (5.14) coincide P (x) is continuous in [0, a]. for x = a − 1. More generally, the function Ssoc P P (x) in [0, a], i.e. we seek a x P We are now to determine the maximum xsoc of Ssoc soc such that P P P1 P2 Ssoc (xsoc ) = max max Ssoc (x), max Ssoc (x) . (5.15) x∈[0,a−1]
x∈[a−1,a]
P 1 (x). Note that We first concentrate on maxx∈[0,a−1] Ssoc P1 P1 max Ssoc max Ssoc (x) = max (i − 1 + q) . x∈[0,a−1]
1≤i≤a−1 q∈[0,1]
(5.16)
P 1 (i − 1 + q) as a function of q. We fix an i ∈ {1, 2, . . . , a − 1} and we consider Ssoc Using elementary analysis we can easily see that there are three cases according to P (i) we have that the function the size of R. More concretely, for R tsoc P 1 Ssoc (i − 1 + q) is respectively decreasing, constant or increasing with respect to q. Hence ⎧ P1 P (i), S (i − 1), if R < tsoc ⎪ ⎪ ⎨ soc P1 P 1 (i − 1 + q), q ∈ [0, 1], if R = t P (i), max Ssoc (i − 1 + q) = Ssoc (5.17) soc ⎪ q∈[0,1] ⎪ ⎩ P1 P if R > tsoc (i). Ssoc (i), P 1 (x) in [0, a − 1] is taken in some In light of (5.17) we have that the maximum of Ssoc integer value of x so (5.16) becomes P1 P1 P1 P1 (x) = max{Ssoc (0), Ssoc (1), . . . , Ssoc (a − 1)}. max Ssoc
x∈[0,a−1]
(5.18)
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P 1 (x) only for integer values of x in [0, a − 1]. So we will now study the function Ssoc (5.13) assumes then the form
P1 Ssoc (i) =
Rμ i−1 )] ρ i+1 C( Rμ C − i) − ρC[ C − (1 + ρ + · · · + ρ , ρ i+1 − 1
i = 0, 1, . . . , a − 1.
(5.19) P 1 (i) is unimodal and so a local maximum We will prove now that the function Ssoc P 1 (i) − S P 1 (i − 1), we of it is also a global maximum. Studying the increments Ssoc soc obtain after some calculations that P1 P1 Ssoc (i) − Ssoc (i − 1) ≥ 0
⇔
P R ≥ tsoc (i),
(5.20)
P (i), i = 1, 2, . . . , a − 1 are given by (5.11). The sequence t P (i) for where tsoc soc i = 1, 2, . . . , a − 1 is easily seen to be increasing with respect to i which shows P 1 (i). Moreover, regarding the maximum i P 1 of S P 1 (i) in the unimodality of Ssoc soc soc {0, 1, . . . , a − 1} we have two cases: P (a − 1) then the maximum i P 1 is characterized by the relationship If R < tsoc soc P P1 P P1 tsoc (isoc ) ≤ R < tsoc (isoc + 1).
(5.21)
P (a − 1) then the maximum i P 1 is attained in a − 1. Hence we obtain If R ≥ tsoc soc ⎧ P1 P (i) ≤ R < t P (i + 1), i = 1, 2, . . . , a − 2, ⎨ Ssoc (i), if tsoc soc P1 max Ssoc (x) = ⎩ S P 1 (a − 1), if R ≥ t P (a − 1). x∈[0,a−1] soc soc (5.22) P 2 (x). Consider the right side of (5.14) as We now concentrate on maxx∈[a−1,a] Ssoc a function of q. Then we have
d P2 (1 − ρ)ρ a Caρ a Ssoc (a − 1 + q) = Rμ − . dq 1 − ρa (1 − qρ a )2
(5.23)
This derivative equals zero for qsoc given by the first branch of (5.12). a) P P2 If qsoc < 0 which happens if and only if R < aC(1−ρ μ(1−ρ) = tsoc (a), then Ssoc (a − P 2 (a − 1 + q) = 1 + q) is decreasing in [0, 1] and therefore we have that maxq∈[0,1] Ssoc P 2 P Ssoc (a − 1). Then by (5.15), (5.19), (5.22) and the continuity of Ssoc (x) (which gives P 2 (a − 1) = S P 1 (a − 1)) we have in particular Ssoc soc P P Ssoc (xsoc ) = max =
max
i∈{0,1,...,a−1}
P1 P2 Ssoc (i), Ssoc (a − 1)
P 1 (i), Ssoc
P (i) ≤ R < t P (i + 1), i = 1, 2, . . . , a − 2, if tsoc soc
P 1 (a − 1), Ssoc
P (a − 1). if R ≥ tsoc
(5.24)
This corresponds to the cases (a)–(b) of the statement of the theorem. Note that in the P (i) for some i = 1, 2, . . . , a − 1, the maximum is taken for all case (a) where R = tsoc x ∈ [i − 1, i] as we have seen in (5.17).
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259
If qsoc ∈ [0, 1) which happens if and only if
aC(1−ρ a ) μ(1−ρ)
P (a) ≤ R < = tsoc
P 2 (a − 1 + q) has a unique maximum at q then Ssoc soc in [0, 1] and thereP 2 P 2 fore we have that maxq∈[0,1] Ssoc (a − 1 + q) = Ssoc (a − 1 + qsoc ). Then by (5.15), (5.19), (5.22) we have that P P P1 P2 Ssoc (xsoc ) = max max Ssoc (i), Ssoc (a − 1) aC μ(1−ρ)(1−ρ a ) ,
i∈{0,1,...,a−1}
P1 P2 (a − 1), Ssoc (a − 1 + qsoc )} = max{Ssoc P2 = Ssoc (a − 1 + qsoc ).
(5.25)
This corresponds to the case (c) of the theorem. aC P2 If qsoc ≥ 1 which happens if and only if R ≥ μ(1−ρ)(1−ρ a ) , then Ssoc (a − 1 + q) is P 2 (a − 1 + q) = S P 2 (a). increasing in [0, 1] and therefore we have that maxq∈[0,1] Ssoc soc As in the other cases we have P P P1 P2 Ssoc (xsoc ) = max max Ssoc (i), Ssoc (a) i∈{0,1,...,a−1}
P1 P2 P2 (a − 1), Ssoc (a)} = Ssoc (a). = max{Ssoc
(5.26)
This corresponds to the case (d) of the theorem. If ρ ≥ 1 then the case (d) never occurs because we have necessarily qsoc < 1. The case (c) becomes (c ). For the case ρ = 1 the reasoning is exactly the ksame but the formulas change a bit because the geometric sums of the type i−1 k=0 ρ are equal to i
i instead of 1−ρ 1−ρ . We can obtain the corresponding results by taking limits as ρ → 1 in the formulas for ρ = 1. Using the results of Theorems 5.2 and 5.3 we have the following figures that show the equilibrium and social optimal thresholds under the P information as R varies. P versus R. (a) Equilibrium pure and mixed strategies: Thresholds xind q 1 1+q • 2C μ
C μ
2
2+q •
2C μ
•
•
•
3C μ
q 1 1+q 2 2+q • • C μ
•
•
•
•
aC μ
a−1
a−1+qind
(a−1)C μ
where qind =
1−
Ca Rμ ρa
.
P a−1+qind xind
•
(a−1)C μ
•
3C μ
a−1
• aC μ
R(ρ≥1)
•
aC μ(1−ρ a )
a
P xind
R(ρ
Optimal balking strategies and pricing for the single server Markovian queue with compartmented waiting space Antonis Economou · Spyridoula Kanta
Received: 24 April 2007 / Revised: 29 July 2008 / Published online: 3 September 2008 © Springer Science+Business Media, LLC 2008
Abstract We consider the single server Markovian queue and we assume that arriving customers decide whether to enter the system or balk based on a natural rewardcost structure, which incorporates their desire for service as well as their unwillingness to wait. We suppose that the waiting space of the system is partitioned in compartments of fixed capacity for a customers. Before making his decision, a customer may or may not know the compartment in which he will enter and/or the position within the compartment in which he will enter. Thus, denoting by n the number of customers found by an arriving customer, he may or may not know n/a + 1 and/or (n mod a) + 1. We examine customers’ behavior under the various levels of information regarding the system state and we identify equilibrium threshold strategies. We also study the corresponding social and profit maximization problems. Keywords Queueing · Balking · Equilibrium strategies · Social optimization · Profit maximization · Nash equilibrium · Pricing · Compartmented waiting space · Partial information Mathematics Subject Classification (2000) 60K25 · 90B22
A. Economou () · S. Kanta Department of Mathematics, Section of Statistics and Operations Research, University of Athens, Panepistemioupolis, Athens 15784, Greece e-mail: [email protected] S. Kanta e-mail: [email protected]
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1 Introduction In the queueing literature there exists an emerging tendency to study systems from an economic viewpoint. More concretely, after studying some performance measures of a system, a certain reward-cost structure is imposed and the objective is the optimization of the system. This control is generally classified as nonadaptive (static) or adaptive (dynamic). In nonadaptive control the objective is to choose the parameters of the system once and for all (e.g. number of servers, servers’ speed, waiting capacity etc.) so that the system behaves optimally. The appropriate mathematical tool for nonadaptive control is the classical optimization theory (non-linear programming). In adaptive control the objective is to choose an optimal policy for the administrator of the system for taking decisions as he observes the evolution of the system (e.g. to accept or not customers, to activate or not a secondary server etc.). In this case, the appropriate mathematical tools lie in the area of Markov decision processes (dynamic programming). Note, however, that in these two types of control, the administrator of the system solves the optimization problem and the customers are forced to follow his solution. If we drop the idea of central planning and we assume that there exists a rewardcost structure for the customers who are allowed to take their own decisions (e.g. to join or balk, to buy priority or not etc.) then the optimization problem can be viewed as a game among the customers. Then the first problem is to find equilibrium strategies for the customers. In a second level, the administrator has to solve the social and the profit maximization problems, taking into account the customers’ behavior. In this case of control, the customers’ behavior is analyzed using a game-theoretic framework and the administrator’s problems are solved by applying classical optimization techniques. In some problems the whole situation should be analyzed as a game between the customers and the administrator. These ideas go back at least to the pioneering works of Naor [24] and Edelson and Hildebrand [10] who studied individual, social and profit maximizing strategies for whether to join or balk in an M/M/1 queue with a simple reward-cost structure. Naor [24] studied the observable case in which each customer knows the number of customers in front of him before his decision, while Edelson and Hildebrand [10] studied the unobservable case. The corresponding results were refined by a number of authors, see e.g. [7, 8, 13, 21, 26]. Over the last years, research on equilibrium, social and profit maximizing strategies in queueing systems has grown considerably. Several authors have investigated such problems in queueing systems incorporating diverse characteristics as priorities, reneging, jockeying, schedules, retrials etc. (see e.g. [3, 9, 12, 17, 19, 23]). Hassin and Haviv [18] summarize the fundamental models and results in this area with extensive bibliographic references. Of particular interest is the value of information available to the customers at their decision epochs. Several papers compare the effect of the information level in the customer strategies and the performance of the system, see e.g. [1, 6, 8, 11, 13, 14, 16, 22]. Recently, the effect of the information level on the balking behavior of the customers has been extensively studied in the framework of call centers, see e.g. [4, 5, 20]. For a recent review of call center theory with many references in the role of information on the balking behavior of the customers see [2], Sect. 3 (demand modulation).
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The study of the balking behavior in the single server Markovian queue (M/M/1 queue) under various levels of information still remains a central problem. Recently, Hassin [15] studied the effect of information and uncertainty on profits in an unobservable M/M/1 queue. He considered scenarios in which the service rate, the service quality, or the waiting conditions are random variables that are known to the server but not to the customers and he asked whether the server is motivated to reveal these parameters. In another direction, Guo and Zipkin [11] studied the M/M/1 queue with a general cost structure under three levels of delay information: no information, partial information (the system occupancy) and full information (the exact waiting time). They identified cases where more accurate delay information improves performance and other cases where a more accurate information implies negative effects for the service provider or the customers. In the present paper we study the balking behavior of the customers in an M/M/1 queue with compartmented waiting space. Again we have a kind of partial information since the customers do not know the exact number of customers that they find upon arrival. Our aim is to complement the existing literature on the effect of information to the balking behavior of the customers by analyzing the effect of the compartmentalization. We present practical motivation for the study of the compartmentalization effect just after the detailed description of the model in Sect. 2. More specifically, we consider the M/M/1 queue and we suppose that the space of the system is partitioned in compartments of fixed capacity for a customers. Each arriving customer decides whether to join in the system or balk. Before making his decision, he may or may not know the compartment in which he will enter and/or the position within the compartment in which he will enter. Let n be the number of customers found by an arriving customer. Then the knowledge of the compartment in which he will enter corresponds mathematically to the knowledge of n/a + 1, while the knowledge of the position within the compartment in which he will enter corresponds to (n mod a)+1. In case where a customer knows both, he knows exactly n and the problem reduces to the fundamental observable model of Naor [24]. If he doesn’t know either, then the problem reduces to the fundamental unobservable model of Edelson and Hildebrand [10]. In this paper we consider separately the cases where all arriving customers observe only n/a + 1 (known compartment number—N case) or only (n mod a) + 1 (known compartment position—P case). We find the equilibrium strategies that are of threshold type. But then the administrator can impose an entrance fee and induce threshold type strategies that maximize the social benefit or his profit. We first show how the various optimal thresholds can be computed efficiently and we provide several qualitative results. We then compare the effect of the information level and of the various parameters of the system on the behavior of the customers and the performance of the system. The paper is organized as follows. In Sect. 2, we describe the parameters and the dynamics of the model, the reward-cost structure and the decision assumptions of the customers. In Sect. 3 we analyze the N case. More specifically we determine equilibrium, social and profit maximizing threshold strategies and we prove a certain ordering for the corresponding thresholds. In Sect. 4 we discuss the qualitative implications of the results and we also present several numerical scenarios. In Sect. 5 we
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carry out the analysis for the P case, while in Sect. 6 we comment on it, by taking a glance to several numerical studies. Finally, in Sect. 7 we summarize the conclusions regarding the influence of the customers’ information on the behavior of the system and we compare the various information cases.
2 The model Customers arrive at a service facility one by one according to a Poisson process with rate λ. There is one server, infinite waiting room and the service times of the customers are assumed to be independent identically distributed exponential random variables with rate μ, independent of the arrival process. We denote by ρ = μλ the traffic intensity of the system. The space of the system is partitioned in compartments of fixed capacity for a customers. The customers are processed on a FCFS basis. The state of the system is described by the process {N (t)} giving the number of customers in system. We are interested in the behavior of the customers when they can decide whether to join or balk upon their arrival. We suppose that there exists a reward of R units for every customer that completes a service in the system. This reward quantifies his satisfaction and/or the added value of being served. There exists also a waiting cost of C units per time unit as long as a customer remains in the system (in queue or in service). This cost quantifies his unwillingness to wait and/or lost benefits. The customers are assumed risk-neutral, i.e. their objective is to maximize the expected value of their net benefit without bothering for its variance. Under the above framework we can think of the situation as a symmetric game among the customers since they are all indistinguishable. Denote the common set of strategies and the payoff function by S and F , respectively. More concretely let F (a, b) be the payoff of a customer that selects strategy a when everyone else selects strategy b. A strategy se is a (symmetric Nash) equilibrium if F (se , se ) ≥ F (s, se ), for every s ∈ S. This means that it is a best response against itself, i.e., if all customers agree to follow it no one can benefit by changing it. A strategy s1 is said to weakly dominate strategy s2 if F (s1 , s) ≥ F (s2 , s), for every s ∈ S and for at least one s the inequality is strict. A strategy s∗ is said to be weakly dominant if it weakly dominates all other strategies in S. In the next two sections we obtain equilibrium customer strategies for joining/balking when the customers have partial information about the number of customers in the system. More concretely, the state-space of the system is written as {0, 1, 2, . . .} = ∞ i=1 Ki , where Ki = {n : (i − 1)a ≤ n < ia} and we consider separately two information cases for the customers: N: Known Compartment Number. Customers observe the number of the compartment in which they are going to enter but not the position within it. More specifically, if there exist n customers in the system just before the arrival of a tagged customer, his information will be the compartment number i = n/a + 1 in which he enters if he decides to join the system.
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241
P: Known Compartment Position. Customers observe the position of the compartment in which they are going to enter but they are not informed about the number of the compartment. The information of an arriving customer is the position i = (n mod a) + 1 in which he enters if he decides to join the system. We finally assume that the decisions of the customers are irrevocable, i.e. retrials of balking customers and reneging of entering customers are not allowed. The compartmentalization of the waiting space described above may reflect the way that the arriving customers observe the system or its real internal structure. As an example of the former case, consider an M/M/1 service system where the customers are forced to wait in a zigzag queue (like the ones in the airports, ticket offices etc.) In such cases the queue folds several times and every fold can be considered as a compartment of approximately a customers. An arriving customer may observe the number of folds in front of him so we have the N case. Alternatively, he may observe just the number of the customers in the last fold of the queue so we have the P case. As an example of the latter case, where the compartmentalization is related to a certain internal structure of the service space, let us consider a system which consists of a different queues and a unique server which operates as follows: The arrivals are assigned in the queues cyclically in a FCFS basis, i.e., the first customer in queue 1, the second customer in queue 2, . . . , the ath customer in queue a, the (a + 1)th customer in queue 1 and so on. The server provides service in a similar way. He serves sequentially the first customer of the queues 1, 2, . . . , a and after the service completion of the first customer in queue a, returns to the first one in queue 1 and so on. This mode of operation preserves the FCFS service discipline of the system and it seems convenient for some service systems where the customers can move very slowly, while the server is very flexible. For example think of a repair facility, where the arriving customers (cars, machines to repair etc.) are assigned to a queues (service places) and the server (repairman) considers the queues cyclically, serving only the first customer of every queue and then proceeding to the next queue. This mode of operation enables the server to use his time more efficiently because he does not have to wait for the upload of the next machine but rather goes to the next queue etc. When he returns in the queue the upload of the next machine to be served has been completed and he can start the service process immediately. In such a system an arriving customer may be informed about the number of items in the queue in which he is assigned. This corresponds to the N case of the model. On the other hand he may be informed just about which queue he will be assigned to and the current position (queue) where the server works. This corresponds to the P case of the model.
3 The known compartment number case: the results Suppose that we are in the framework of the known compartment number (N) case. Then a pure strategy is specified by a set A ⊆ {1, 2, . . .} which shows the ‘favorable’ compartment numbers for a customer, i.e. a customer decides to join the system if he knows that he will enter to a compartment with a number belonging to A. First, we are interested in conditions on the parameters of the model that ensure that the system
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is non-degenerate, i.e. some customers are willing to enter and the system is not continuously empty. Second, we seek for equilibrium pure strategies. We summarize the relevant results in the following. Theorem 3.1 The compartmented M/M/1 queue under the N information is nondegenerate if and only if 1 1 Rμ + ≥ 1, − aC 1 − ρ a a(1 − ρ) Rμ a − 1 + ≥ 1, aC 2a
if ρ = 1,
if ρ = 1.
(3.1) (3.2)
There exists a unique dominant pure strategy (which is also equilibrium). It is the threshold strategy ‘Enter if the compartment number that you will be assigned is less N = x N ’, with than or equal to iind ind Rμ N xind
=
aC Rμ aC
+
1 1−ρ a
+
a−1 2a
−
1 a(1−ρ)
if ρ = 1, if ρ = 1.
(3.3)
Proof Consider a tagged customer and suppose that all other customers follow a strategy A. Then the Markov chain describing the number of customers in the system will be eventually absorbed in the set {0, 1, 2, . . . , i ∗ a}, where i ∗ is the maximum integer such that {0, 1, 2, . . . , i ∗ } ⊆ A. Indeed, a moment of reflection shows that under strategy A all other states become transient and the system behaves as an M/M/1/i ∗ a queue. As it is well known, the stationary distribution of this queue is given by ⎧ n ⎨ (1−ρ)ρ , n = 0, 1, . . . , i ∗ a, if ρ = 1 ∗ 1−ρ i a+1 (3.4) P [Q = n] = ⎩ 1 , n = 0, 1, . . . , i ∗ a, if ρ = 1. ∗ i a+1 Consider now the tagged customer who is to decide whether to join the system or not, given the information that he can enter in the compartment number i. So he knows that the number of customers at his arrival is n ∈ {(i − 1)a, (i − 1)a + 1, . . . , ia − 1}. If he decides to enter then his expected net individual benefit is N Sind (i) = R −
C {E[Q− |Q− ∈ {(i − 1)a, (i − 1)a + 1, . . . , ia − 1}] + 1}, μ
(3.5)
where Q− is a random variable having the equilibrium distribution of the number of customers at an arrival instant in the M/M/1/i ∗ a queue. Because of the PASTA property we have that the distribution of Q− coincides with that of Q in (3.4). Using (3.4) in (3.5) we obtain that N Sind (i) =
R− R−
C μ {ia C μ {ia
− −
a 1 1−ρ a + 1−ρ }, a−1 2 },
if ρ = 1, if ρ = 1.
(3.6)
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243
N (i) ≥ 0, that is when The tagged customer will decide to enter if and only if Sind Rμ 1 1 a−1 i ≤ Rμ aC + 1−ρ a − a(1−ρ) , for ρ = 1 or when i ≤ aC + 2a , for ρ = 1. Hence, the N = x N with x N given by (3.3) is a dominant threshold strategy with threshold iind ind ind strategy. To ensure that the system does not remain for ever empty we must have N = 0 which gives the conditions (3.1) and (3.2). iind
We now consider the problem of social and profit maximization. By imposing an admission fee, the administrator of the system can force the customers to adopt any threshold i he may desire. Indeed, for a desired i he must set the admission fee p such that (R−p)μ 1 1 aC + 1−ρ if ρ = 1, a − a(1−ρ) i= (3.7) (R−p)μ aC + a−1 if ρ = 1. 2a N that maximizes the mean social net profit per time unit. We seek for a threshold isoc This is given in the next theorem.
Theorem 3.2 In the compartmented M/M/1 queue under the N information, the threshold strategy that maximizes the social net profit per time unit is ‘Enter, if the N ’, where compartment number that you will be assigned is less than or equal to isoc N N = x N , x N being the unique solution of the equation g(x) = x isoc soc soc ind in [1, ∞] with ⎧ xa+1 ) 1 1 ⎨ (xa+1)(1−ρ a )−a(1−ρ + 1−ρ if ρ = 1 a − a(1−ρ) a(1−ρ)(1−ρ a ) g(x) = (3.8) ⎩ a x 2 − a−2 x if ρ = 1. 2
2
Moreover the optimal social threshold is smaller than or equal to the optimal individual threshold, i.e. N N ≤ iind . isoc
(3.9)
N (i) = λ∗ R − Proof For any threshold i, the social net profit per time unit is Ssoc ∗ CE[Q], where λ is the mean arrival rate and E[Q] the mean number of customers in system, given that the customers follow the threshold policy with threshold i. In this case the system behaves as an M/M/1/ia queue and using the stationary distribution given by (3.4) we obtain
N (i) = Ssoc
⎧ ⎨ λR
1−ρ ia ρ − C{ 1−ρ 1−ρ ia+1 ⎩ λR ia − C ia , ia+1 2
−
(ia+1)ρ ia+1 }, 1−ρ ia+1
if ρ = 1, if ρ = 1.
(3.10)
N (i) is unimodal with Suppose now that ρ = 1. We will prove that the function Ssoc N respect to i and that isoc as defined in the statement of the theorem is a local maximum N (i); hence a global maximum. To this end we study the increments S N (i) − of Ssoc soc
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N (i − 1). After some long but tedious calculations we obtain that Ssoc N N (i) − Ssoc (i − 1) ≥ 0 Ssoc
⇔
(ia + 1)(1 − ρ a ) − a(1 − ρ ia+1 ) Rμ ≤ a(1 − ρ)(1 − ρ a ) aC
⇔
N , g(i) ≤ xind
(3.11)
N are given by the first branches of (3.8) and (3.3) respectively. To where g(i) and xind N (i), it is now sufficient to prove that g(x) is increasing. prove the unimodality of Ssoc To this end, define the function f : [1, ∞) → R with f (x) = g(x) − x. We have ρ ρ f (x) = 1−ρ + (1−ρaaln)(1−ρ) ρ xa+1 and we can easily see that f (x) ≥ 0 for x ≥ 1. Hence we have that g(x) is increasing in [1, ∞) and limx→∞ g(x) = ∞. Note also N (i) is unimodal. In light of (3.11), its global maximum i N is that g(1) = 1. So Ssoc soc characterized by N N N ) ≤ xind ≤ g(isoc + 1). g(isoc
(3.12)
N = x N , where x N is the unique solution of the equation Hence we have that isoc soc soc N . g(x) = xind It remains now to prove (3.9). Because f (x) is increasing in [1, ∞) and f (1) = N ) ≥ 0, i.e. g(x N ) − x N = x N − x N ≥ g(1) − 1 = 0 we have in particular that f (xsoc soc soc soc ind N ≤ iN . 0. But then taking floors we have isoc ind The case where ρ = 1 is done similarly using the second branches of (3.3), (3.8) and (3.10).
We now proceed to the profit maximization problem. Here, the administrator of the system imposes an entrance fee for maximizing his own benefit per time unit. In the following theorem we provide the profit maximization threshold. Theorem 3.3 In the compartmented M/M/1 queue under the N information, the threshold strategy that maximizes the administrator’s net profit per time unit is ‘Enter, N ’, if the compartment number that you will be assigned is less than or equal to iprof N = x N , x N being the unique solution of the equation h(x) = x N in where iprof prof prof ind [1, ∞] with xa−a )(1−ρ xa+1 ) x + (1−ρ , if ρ = 1, ρ xa−a (1−ρ)(1−ρ a ) (3.13) h(x) = x + (x − 1)(xa + 1), if ρ = 1. Moreover the optimal profit maximization threshold is smaller than or equal to the optimal social threshold, i.e. N N iprof ≤ isoc .
(3.14)
Proof In light of (3.7) the maximum entrance fee that can be imposed by the administrator of the system in order to force the customers to adopt a given threshold i is ⎧ 1 1 ⎨ R − aC μ (i − 1−ρ a + a(1−ρ) ), if ρ = 1, p(i) = (3.15) ⎩ R − aC (i − a−1 ), if ρ = 1. μ 2a
Queueing Syst (2008) 59: 237–269
245
N (i) = λ∗ p(i) where λ∗ is the mean Then his expected profit per time unit is Sprof arrival rate given that the customers follow the threshold i. The system behaves as an M/M/1/ia queue and therefore using (3.4) we obtain after some algebra that
N Sprof (i) =
⎧ ia ) N ⎨ λaC(1−ρ ia+1 (xind − i), μ(1−ρ
⎩
)
λa 2 Ci N μ(ia+1) (xind
− i),
if ρ = 1, (3.16) if ρ = 1,
N is given by (3.3). where xind Suppose now that ρ = 1. Then, after some calculations we obtain that N (i) Sprof N (i − 1) Sprof
≥1
⇔
N h(i) ≤ xind ,
(3.17)
N are given by the first branches of (3.13) and (3.3) respecwhere h(i) and xind tively. The function h(x) is easily seen to be increasing in [1, ∞), h(1) = 1 and N (i) is unimodal and its maximum i N is characterized by limx→∞ h(x) = ∞ so Sprof prof N N N h(iprof ) ≤ xind ≤ h(iprof + 1).
(3.18)
N = The function h(x) is continuous so we have that the optimal threshold is iprof N , where x N is the unique solution of h(x) = x N in [1, ∞). xprof prof ind N (i) < 0 We now proceed to prove (3.14). From (3.16) we have obviously that Sprof N so we have immediately i N ≤ i N . We have also established that i N ≤ for i > iind soc prof ind N in (3.9). Therefore in the case where i N = i N we have immediately (3.14). iind soc ind N < i N . It suffices to show that for integers i with i N ≤ i < i N , Suppose that isoc soc ind ind N N (i), i.e. S N (i) is decreasing in {i N , i N + 1, . . . , i N }. we have Sprof (i + 1) ≤ Sprof soc soc prof ind N (i), we will have then (3.14). Because of the unimodality of Sprof N , we have from the monotonicity of g(x) given in (3.8) and from (3.12) For i ≥ isoc N . To prove that S N (i + 1) ≤ S N (i) we have to prove in light of that g(i + 1) ≥ xind prof prof N . So it suffices to prove that h(i + 1) ≥ g(i + 1), i.e. (3.17) that h(i + 1) ≥ xind
i
1 ρ a (1 − ρ ia+1 ) 1 (1 − ρ ia+1 )(1 − ρ ia+a ) − + . ≤ i + 1 − ρ (1 − ρ)(1 − ρ a ) 1 − ρ a ρ ia (1 − ρ)(1 − ρ a )
(3.19)
This reduces after some simplification to prove i
1 − ρ ia ρ ≤ ia . 1 − ρ ρ (1 − ρ)(1 − ρ a )
(3.20)
For ρ < 1, (3.20) is written equivalently as i≤
1 1 1 1 − ρ ia = + + · · · + ia+1 , ρ ia+1 (1 − ρ a ) ρ a+1 ρ 2a+1 ρ
(3.21)
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Queueing Syst (2008) 59: 237–269
which is clearly valid. For ρ > 1, (3.20) is written similarly as i≥
1 1 − ρ ia 1 1 = + + · · · + ia+1 , ρ ia+1 (1 − ρ a ) ρ a+1 ρ 2a+1 ρ
(3.22)
which is again valid. The case where ρ = 1 is treated similarly using the second branches of (3.3), (3.13) and (3.16). By solving (3.3), (3.12) and (3.18) with respect to R and using the results of Theorems 3.1, 3.2 and 3.3 we have the following figures that show the equilibrium, social optimal and profit maximizing thresholds under the N information as R varies. N versus R. (a) Equilibrium pure strategies: Thresholds iind
1
N (1) tind
2
• N (2) tind
3
•
•
•
•
•
N (3) tind
where N tind (i) =
N iind
R
⎧ ⎨C μ {ai −
a 1−ρ a
⎩ aC (i −
a−1 2a ),
μ
+
1 1−ρ },
ρ = 1, ρ = 1.
N versus R. (b) Social Optimization: Thresholds isoc
1
N (1) tsoc
2
• N (2) tsoc
•
3
•
•
•
•
N (3) tsoc
where N (i) = tsoc
N isoc
R
⎧ a ia+1 ) ⎨ C (ia+1)(1−ρ )−a(1−ρ , a
ρ = 1,
⎩ aC {ai 2 − (a − 2)i −
ρ = 1.
μ
(1−ρ)(1−ρ )
2μ
a−1 a },
N versus R. (c) Profit Maximization: Thresholds iprof
1
N (1) tprof
• N (2) tprof
2
• N (3) tprof
3
N iprof
•
•
•
• R
Queueing Syst (2008) 59: 237–269
where
⎧ ⎨ aC {i +
μ N tprof (i) = ⎩ aC {i μ
247
(1−ρ (i−1)a )(1−ρ ia+1 ) ρ (i−1)a (1−ρ)(1−ρ a )
+ (i − 1)(ia + 1) −
−
1 1−ρ a
a−1 2a },
+
1 a(1−ρ) },
ρ = 1, ρ = 1.
4 The known compartment number case: discussion In this section we discuss some qualitative properties and implications of the results that concern the N case. Moreover, we present some numerical results to illustrate the dependence of the optimal thresholds on the parameters of the model. 4.1 The effect of the externalities The first remark concerns the discrepancy between individual and social optimization in the N case. Indeed the relation (3.9) shows that individual optimization leads to longer queues than are socially desired. This happens because an arriving customer that decides to enter the system ignores the negative externalities that imposes on future arrivals. Indeed such a customer may join the queue even if his own expected benefit is smaller that the expected reduction in the benefit of future customers. The same situation occurs in [24] model. Hassin [13] described an equivalent model operating under the LCFS-PR discipline, where the customers do not impose externalities and they behave socially optimally. This result shows that the externalities constitute the unique source for the discrepancy between the socially and the individually optimal behavior of the customers. We will now show that the same reasoning can be extended in the framework of our model. More specifically we show that in an equivalent model with the LCFS-PR discipline the customers do not impose externalities on the others and their individual optimal threshold coincides with the social optimal threshold given in Theorem 3.2. More concretely, we suppose that newly arriving customers join the system and start immediately being served, possibly preempting the service of another customer. Moreover, when a preempted customer reenters for service, his service is continued from the point of interruption. Regarding the information, we suppose that the customers know continuously the compartment in which they reside but not the position within it (N case). The relevant decision that an individual faces is when to leave the system. Because of the memoryless property of the exponential distribution and the fact that a customer knows continuously only the compartment in which he resides, a moment of reflection shows that the customer has incentive to leave the system only at epochs when he is moved to another compartment because of a new arrival. If the customers follow a threshold strategy i ∗ , i.e. they do not abandon the system as long as they reside to a compartment less than or equal to i ∗ , then the system behaves as an M/M/1/i ∗ a queue. Moreover, because of the LCFS-PR discipline the customers do not impose externalities to other customers. Indeed, if we consider a tagged customer, it is clear he does not impose externalities to future customers, because all of them are placed in front of him in the queue. On the other hand, by the memoryless property of the exponential distribution and the homogeneity of the
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customers, we can see that the tagged customer may decide to renege, only when there are no customers behind him. For, if it is beneficial for the tagged customer to renege at some time, then a customer who is behind him at that time should have been removed earlier, when he was at the same position as the tagged customer now. Therefore, when a customer decides whether to renege or not, he imposes no externalities whatever he does, because everybody present is served prior to him as well as the future arrivals. We now determine the individually optimal policy for a customer under the LCFSN −LCFSPR PR discipline. Let Sind (i) be the expected net benefit of the last customer in the system (this is the only that can renege) given that he has just moved to the compartment i because of a new arrival and given that all customers renege when they enter the compartment i + 1 (including the customer under consideration). He then knows that he is exactly at the position (i − 1)a + 1 of the system (first in the compartment i). The customer prefers to stay in the system for all compartment N −LCFSPR numbers i such that Sind (i) ≥ 0. We have that N −LCFSPR Sind (i) = Rfi − CSi ,
(4.1)
where fi is the probability that a customer will reach state 0 (completion of service) before reaching state ia + 1 (abandonment), starting from the position (i − 1)a + 1 and Si is the corresponding mean remaining time till he leaves the system (either because of service completion or abandonment). The probability fi is the classical gambler-ruin probability, where the initial point is (i − 1)a + 1, the goal is ia + 1 λ and the winning probability in each round is p = λ+μ and q = 1 − p. Moreover, Si 1 is the expected number of rounds till the game is over multiplied by λ+μ (the mean time till the next transition). Using the well known formulas for these quantities (see e.g. [25], Sect. 4.4, Example 4.4(a)) we conclude that ⎧ a ⎨ 1−ρia+1 , if ρ = 1, 1−ρ fi = (4.2) ⎩ a , if ρ = 1, ia+1
and Si =
⎧ ⎨ ⎩
1 μ(1−ρ) {(ia a 2μ {(i
a
1−ρ + 1) 1−ρ ia+1 − a},
− 1)a + 1},
if ρ = 1, if ρ = 1.
(4.3)
N −LCFSPR By plugging (4.2) and (4.3) in (4.1) and solving Sind (i) ≥ 0 for i we arrive after some straightforward calculations at (3.11). Hence we conclude that the individually optimal policy in this case coincides with the socially optimal policy of the initial model. Therefore, the externalities constitute the unique source for the discrepancy between the socially and the individually optimal behavior of the customers.
4.2 Dependence of the optimal threshold on the model parameters In this section we present some numerical results to illustrate the behavior of the optimal thresholds iind , isoc , iprof for the N case when keeping all parameters fixed and let only one vary. In all applications we consider the waiting cost C per time
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249
Fig. 1 Optimal thresholds with respect to R − N case
unit and the service rate μ to be equal to 1. This can be always assumed, without loss of generality, with appropriate re-scaling of the time and reward/cost units. So we first examine the behavior of the thresholds as R or λ varies. Then, when the compartment size a varies, we depict iind a, isoc a, iprof a (the maximum numbers of potential customers under the optimal policies) as functions of a. This is clearly more informative than depicting iind , isoc , iprof with respect to a. In Fig. 1 we present the equilibrium, social optimization and profit maximization thresholds with respect to R for λ = 0.7 and a = 4. As we can observe the three thresholds are all increasing ladder functions of R. The individual optimal threshold increases more rapidly than the other optimal thresholds. This is because of the form N (i) which are linear in i. The profit maximizing threshold increases very slowly of tind with respect to R. This fact can be explained due to the form of the critical numbers N (i) which increase exponentially fast. tprof Figure 2 shows the equilibrium, social optimization and profit maximization thresholds with respect to λ for R = 50 and a = 4. Note that the equilibrium threshold converges to Rμ aC as λ increases (because of (3.3)). The social optimization and profit maximization thresholds are unimodal functions of λ and they become ultimately 1 for large values of λ. In Fig. 3 we display the maximum number of potential customers for the corresponding equilibrium, social optimization and profit maximization thresholds with respect to a for λ = 0.7 and R = 50. As a increases above a certain level all thresholds become equal to 1. This justifies the linear part at the right part of the figure. In the meanwhile the thresholds are not monotone functions of a. For example in the curve of the equilibrium threshold we see that there exist many local maxima and minima. N a is near to the full information case equiIn particular we see that the minimum iind librium threshold (a = 1). The same value is also attained for a = 2, 5, 10, 25, 50, i.e.
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Fig. 2 Optimal thresholds with respect to λ − N case
Fig. 3 Maximum number of potential customers with respect to a − N case
for all values of a that are divisors of the full information case equilibrium threshold. This explains the non-monotone behavior of the optimal threshold with respect to a. The maximum number of potential customers in the system under the full information is 50. We expect that for other values of a the corresponding optimal threshold N a is also near 50. So when a = 20 we would expect the threshshould be such that iind
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251
Fig. 4 Optimal social benefit and administrator’s profit with respect to a − N case
old to be equal to 2 or 3. As we can observe the threshold is 3, letting 60 customers to enter the system. On the other hand for a = 25 the adequate threshold would drop to 2 so as to be in agreement with the full information case. This is exactly what happens. That difference between the two cases of a is due to the partial information that the customers receive in the N case. Same observations occur with the social and profit maximizing thresholds. The message for the designer of the system is that the compartmentalization should ‘agree’ with the optimal threshold if the customers had full information. The compartment size a should be chosen close to some divisor of the optimal threshold for the corresponding fully observable model. In the last Fig. 4, we present the social benefit and administrator’s profit, when the customers follow the corresponding optimal policies, with respect to a for λ = 0.9 and R = 25. For a = 1 which corresponds to full information of the customers about the state of the system, the optimal social benefit is high, while the administrator’s profit attains its minimum value. As we can observe for small values of a the difference of the two functions is positive, whereas for greater values of a the two functions coincide. Note that there is a value of a, in our example this value is a = 7, such that the administrator’s profit is maximized and then it decreases. This is in some sense the ‘ideal’ compartment size for the administrator. Moreover, in that case the whole social benefit is taken by the administrator. Thus, we conclude that the administrator can improve its profit by an adequate selection of the compartment size. 4.3 Comparison with Naor’s observable model A possible interpretation for the N case is that the customers take their decisions whether to join or to balk, having only a rough information about the number of customers in the system. For example, for a = 12 or a = 100 the customers count
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how many dozens or hundreds of customers are in the system. So, it is reasonable for the form of the optimal policies in the N case to be similar to the form of the optimal policies in Naor’s [24] observable model as it is confirmed by Theorems 3.1, 3.2 and 3.3. On the other hand, the optimal thresholds in the N case are significantly different than the optimal thresholds of Naor’s [24] model. In Naor’s [24] model, the equilibrium threshold Rμ C does not depend on λ while the equilibrium threshold of the N case given in Theorem 3.1 clearly does. This is because the arrival rate does not offer any information to the customer in the observable model. However, in the N case, it influences customer’s estimate about his position in the queue (see (3.4) and (3.5)). The numerical studies for Naor’s [24] model regarding the effect of the arrival rate on the thresholds have been investigated by Hassin [13]. In the N case of our model we have that the optimal equilibrium threshold is decreasing in λ. For the social and profit maximizing thresholds the numerical results agree with the findings of Hassin [13] for Naor’s model, i.e. both thresholds are unimodal functions of the arrival rate. A noteworthy similarity with Naor’s [24] model is the existence of a dominant strategy in the N case. The existence of dominant strategies is typical only for fully observable models (see e.g. [6, 24]). In general, although equilibrium strategies do exist in partially observable models, dominant strategies do not exist. Indeed, in partially observable models, the customer’s net benefit function Sind (i) depends both on the state of the system and the strategy of the other customers. However, in the N case N (i) in (3.5) does not depend on the of the present model we have that the function Sind ∗ strategy i of the other customers. This happens because the conditional distribution of the queue length Q− given that Q− ∈ {(i − 1)a, (i − 1)a + 1, . . . , ia − 1} in an M/M/1/i ∗ a queue is the same for any i ∗ ≥ i. So the N case of the present model is a rare example where a partially observable model has a dominant strategy.
5 The known compartment position case: the results We now turn to the framework of the known compartment position (P) case. A pure strategy here is specified by a set A ⊆ {1, 2, . . . , a} which shows the favorable compartment positions for a customer, i.e. in this case a customer decides to join the system if and only if he will take a position belonging to A. Note here that the number of pure strategies is finite and equals 2a . We consider also mixed strategies. A mixed strategy is specified by a vector (q1 , q2 , . . . , qa ), where qi is the probability that a customer enters the system when he is given the information that he will be assigned to position i of the compartment. A mixed strategy with qi = 1 for 1 ≤ i ≤ j , qj +1 = q and qi = 0 for j + 1 < i ≤ a is referred to as a mixed threshold strategy with threshold j + q. In the special case where q = 0 we have a pure threshold strategy with threshold j . We establish conditions that ensure that the system is non-degenerate and we also determine the equilibrium mixed strategies. To this end we begin by studying the equilibrium behavior of the system given that all customers follow the same mixed strategy (q1 , q2 , . . . , qa ). Note that there are two cases that should be treated differently. The first case refers to strategies with qi = 0 for some i. In this case the Markov chain that describes the number of customers in the system is eventually
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253
Fig. 5 Transition diagram—P case (a)
Fig. 6 Transition diagram—P case (b)
absorbed to the finite set {0, 1, . . . , i ∗ }, where i ∗ + 1 is the first i with qi = 0, i.e. q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0. The corresponding transition diagram is given in Fig. 5. The second case covers strategies with qi > 0 for all i = 1, 2, . . . , a. Then the Markov chain is irreducible in the set N0 of the non-negative integers and the transition diagram is given in Fig. 6. Using the standard product-form formula for the stationary distribution of birthdeath processes we have the following. Lemma 5.1 Consider the compartmented M/M/1 queue under the P information. (a) If all customers follow a strategy (q1 , q2 , . . . , qa ) with q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0 for some i ∗ = 1, 2, . . . a − 1, then the queue is always stable and its stationary distribution is given by B, n = 0, P [Q = n] = (5.1) Bρ n q1 q2 . . . qn , n = 1, 2, . . . i ∗ , where
⎛
⎞−1
∗
B = ⎝1 +
i
ρ n q 1 q 2 . . . qn ⎠
.
(5.2)
n=1
(b) If all customers follow a strategy (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a then the queue is stable if and only if q1 q2 . . . qa ρ a < 1.
(5.3)
Then its stationary distribution is given by P [Q = n] = Bρ n (q1 q2 . . . qa )n/a
(n mod
a)
qj ,
n = 0, 1, . . . ,
(5.4)
j =1
where
a t −1 t=1 q1 q2 . . . qt ρ B = 1+ 1 − q 1 q 2 . . . qa ρ a
(where the product in (5.4) is interpreted as 1 for n ≡ 0 mod a).
(5.5)
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We are now in position to find all equilibrium strategies in the P case. We have the following. Theorem 5.2 The compartmented M/M/1 queue under the P information is nondegenerate if and only if R≥
C . μ
(5.6)
The best responses against any given strategy are always of threshold type (pure or mixed). Hence, the equilibrium strategies are of threshold type. More concretely, if ρ < 1, we have the following cases as R varies in [ C μ , ∞): (a) For R = i C μ for some i = 1, 2, . . . , a − 1 there exist multiple equilibrium strategies. They are the mixed threshold strategies with thresholds in [i − 1, i]. C (b) For R ∈ (i C μ , (i + 1) μ ) for some i = 1, 2, . . . , a − 1 there exists a unique equilibrium strategy. It is the pure threshold strategy with threshold i. C (c) For R ∈ [a C μ , a μ(1−ρ a ) ) there exists a unique equilibrium strategy. It is the threshold strategy with threshold a − 1 + (1 −
Ca −a Rμ )ρ .
C (d) For R ∈ [a μ(1−ρ a ) , ∞) there exists a unique equilibrium strategy. It is the pure threshold strategy with threshold a. If ρ ≥ 1, we still have the cases (a) and (b) as above, but instead of the cases (c) and (d) we have the case (c ) For R ∈ [a C μ , ∞) there exists a unique equilibrium strategy. It is the threshold
strategy with threshold a − 1 + (1 −
Ca −a Rμ )ρ .
Proof We are interested in finding the best responses for every given mixed strategy (q1 , q2 , . . . , qa ). To this end we consider a tagged customer and suppose that all other customers follow a strategy (q1 , q2 , . . . , qa ). We have to distinguish two cases because the stationary distribution of the number of customers has two different forms corresponding to the cases (a) and (b) of Lemma 5.1. We first consider the case where all customers follow a strategy (q1 , q2 , . . . , qa ) with q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0 for some i ∗ = 1, 2, . . . , a − 1. Now the system is always stable (see Lemma 5.1). We consider a tagged customer who is to decide whether to join or balk, given the information that he can enter in the position i of the compartment. So he knows that the number of customers at his arrival is exactly n = i − 1 and his expected net individual benefit is P (i) = R − Sind
C i. μ
(5.7)
P (i) is decreasing in i so we have that there exists an i such that The function Sind P P (i) < 0 for i = i + 1, . . . , a. Hence a best reSind (i) ≥ 0 for i = 0, 1, . . . , i and Sind P (i ) = 0 then every mixed threshold strategy
sponse is the threshold strategy i . If Sind P (i ) > 0 the with threshold i ∈ [i − 1, i ] is best response, while in the case with Sind
threshold strategy i is the unique best response against (q1 , q2 , . . . , qa ).
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255
We conclude that among strategies (q1 , q2 , . . . , qa ) with q1 , q2 , . . . , qi ∗ > 0 but qi ∗ +1 = 0 for some i ∗ = 1, 2, . . . a − 1, the only possible equilibrium is of the form (1, 1, . . . , 1, q, 0, 0, . . . , 0) with q ∈ (0, 1], i.e. mixed threshold strategies with thresholds in (0, a − 1]. Note now that a strategy (1, 1, . . . , 1, q, 0, 0, . . . , 0) with q ∈ (0, 1) P (i) = 0 which gives R = i C in the ith position can be equilibrium if and only if Sind μ for some i = 1, 2, . . . , a − 1. This corresponds to case (a) of the theorem. Similarly, a strategy (1, 1, . . . , 1, 0, 0, . . . , 0) with 1s up to the ith position is unique equilibrium P (i) > 0 and S P (i + 1) < 0. Solving for R we obtain case (b) of the if and only if Sind ind theorem. We now consider the other case of Lemma 5.1 and suppose that all customers follow a strategy (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a. If the system is unstable under this strategy, i.e. q1 q2 . . . qa ρ a ≥ 1 then the best response of the tagged customer is not to enter. Hence, in case that the condition (5.3) does not hold a strategy (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a cannot be an equilibrium. Suppose now that the condition (5.3) holds and the system is stable and consider a tagged customer who is to decide whether to join or balk, given the information that he can enter in the position i of the compartment. So he knows that the number of customers at his arrival is n ∈ {i − 1, a + i − 1, 2a + i − 1, . . .}. If he decides to enter then his expected net individual benefit is P (i) = R − Sind
C {E[Q− |Q− ∈ {i − 1, a + i − 1, 2a + i − 1, . . .}] + 1}, μ
(5.8)
where Q− is a random variable having the equilibrium distribution of the number of customers at an arrival instant. Because of PASTA property we have that Q− is distributed as Q in (5.4). Using (5.4) we compute the expected value in (5.8) and we obtain that the expected net individual benefit is C ρ a q 1 q 2 . . . qa P (i) = R − + i . (5.9) Sind a μ 1 − ρ a q1 q2 . . . qa P (i) is decreasing in i so we have that there exists an i such that The function Sind P (i) ≥ 0 for i = 0, 1, . . . , i and S P (i) < 0 for i = i + 1, . . . , a. Hence a best reSind ind P (i ) = 0 then every mixed threshold strategy sponse is the threshold strategy i . If Sind P (i ) > 0 the threshold strategy i
with threshold i ∈ [i − 1, i ] is best response. If Sind is the unique best response against (q1 , q2 , . . . , qa ). We conclude that among strategies (q1 , q2 , . . . , qa ) with qi > 0 for all i = 1, 2, . . . , a, the only possible equilibrium is of the form (1, 1, . . . , q) with q ∈ (0, 1], i.e. mixed threshold strategies with thresholds in (a − 1, a]. Note that under a policy (1, 1, . . . , q), (5.9) assumes the form ρa q C P a +i . (5.10) Sind (i) = R − μ 1 − ρa q P (a − 1) > 0 and S P (a) = 0 for the case To be an equilibrium we should have Sind ind P P (a − 1) ≥ 0 for R and then q ∈ (0, 1) or Sind (a) ≥ 0 for the case q = 1. Solving Sind C Ca P (a) = 0 for q gives R ∈ [a C , a −a which is the case Sind μ μ(1−ρ a ) ) and q = (1 − Rμ )ρ
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P (a) > 0 for R gives the case (d) of the theorem. Of (c) of the theorem. Solving Sind course for ρ ≥ 1 we should have q < ρ −a for stability and the case (d) does not occur. The case (c) becomes then (c ).
We can now proceed to the problem of social and profit maximization. Again by imposing an appropriate admission fee, the administrator of the system can oblige the customers to adopt any desired threshold i. So we seek threshold strategies that maximize the social net reward and the administrator’s profit per time unit. Theorem 5.3 In the compartmented M/M/1 queue under the P information define P (i), i = 1, 2, . . . , a by the numbers tsoc
P tsoc (i) =
⎧ C ⎪ ⎪ μ(1−ρ) {i − ⎪ ⎪ ⎪ ⎪ aC(1−ρ a ) ⎪ ⎨ μ(1−ρ) , i(i+1)C ⎪ ⎪ ⎪ 2μ , ⎪ ⎪ ⎪ ⎪ ⎩ a2 C μ ,
ρ(1−ρ i ) 1−ρ },
i = 1, 2, . . . , a − 1, ρ = 1, i = a, ρ = 1, i = 1, 2, . . . , a − 1, ρ = 1,
(5.11)
i = a, ρ = 1
and let qsoc defined by ⎧ a) ⎪ ⎨ ρ1a 1 − Ca(1−ρ Rμ(1−ρ) , qsoc = ⎪ ⎩ 1 − Ca 2 , Rμ
ρ = 1, (5.12) ρ = 1.
If ρ < 1, the threshold strategies that maximize the social net reward per time unit as R varies in [ C μ , ∞) are given as follows: P (i) for some i = 1, 2, . . . , a − 1 there exist multiple social optimal (a) For R = tsoc threshold strategies with thresholds in [i − 1, i]. P (i), t P (i + 1)) for some i = 1, 2, . . . , a − 1 there exists a unique so(b) For R ∈ (tsoc soc cial optimal threshold strategy. It is the pure threshold strategy with threshold i. aC P (a), (c) For R ∈ [tsoc μ(1−ρ)(1−ρ a ) ) there exists a unique social optimal threshold
strategy. It is the threshold strategy with threshold a − 1 + qsoc . aC (d) For R ∈ [ μ(1−ρ)(1−ρ a ) , ∞) there exists a unique social optimal threshold strategy. It is the pure threshold strategy with threshold a. If ρ ≥ 1, we still have the cases (a) and (b) as above, but instead of the cases (c) and (d) we have the case P (a), ∞) there exists a unique social optimal threshold strategy. It is (c ) For R ∈ [tsoc the threshold strategy with threshold a − 1 + qsoc . P (x) to be the social net profit per Proof For a given threshold x ∈ [0, a] define Ssoc ∗ time unit. Letting λ (x) and E[Q](x) be the mean arrival rate and the mean number of customers respectively, given that the customers follow the threshold policy x, we P (x) = λ∗ (x)R − CE[Q](x). Of course S P (0) = 0. have that Ssoc soc
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257
Suppose that ρ < 1. If x ∈ (0, a − 1] then x = i − 1 + q with i ∈ {1, 2, . . . , a − 1} and q ∈ (0, 1]. The stationary distribution of the system is given by (5.1) and so i−1 and E[Q](x) = we can compute after some algebra that λ∗ (x) = λ 1−(1−q(1−ρ))ρ 1−(1−q(1−ρ))ρ i ρ(1−ρ i )−i(1−q(1−ρ))ρ i (1−ρ) . (1−ρ)(1−(1−q(1−ρ))ρ i ) P1 Ssoc (x) =
P (x) assumes the form Then the function Ssoc
i−1 )] − (1 − q(1 − ρ))ρ i C[ Rμ − i] ρC[ Rμ C − (1 + ρ + · · · + ρ C , 1 − (1 − q(1 − ρ))ρ i
x = i − 1 + q, i ∈ {1, 2, . . . , a − 1}, q ∈ (0, 1].
(5.13)
Note that for x = 0, (5.13) is also valid for i = 1 and q = 0. If x ∈ (a − 1, a] then x = a − 1 + q with q ∈ (0, 1]. The stationary distribua−1 and tion of the system is given by (5.4) and we obtain λ∗ (x) = λ 1−(1−q(1−ρ))ρ 1−ρ a E[Q](x) =
aqρ a 1−qρ a
+
ρ 1−ρ
−
aρ a 1−ρ a .
P (x) assumes the form In this case the function Ssoc
1 − (1 − q(1 − ρ))ρ P2 (x) = λR Ssoc a
a−1
1−ρ
aqρ a ρ aρ a , −C + − 1 − qρ a 1 − ρ 1 − ρ a
x = a − 1 + q, q ∈ (0, 1].
(5.14)
Note that for x = a − 1 (5.14) is also valid for q = 0, that is (5.13) and (5.14) coincide P (x) is continuous in [0, a]. for x = a − 1. More generally, the function Ssoc P P (x) in [0, a], i.e. we seek a x P We are now to determine the maximum xsoc of Ssoc soc such that P P P1 P2 Ssoc (xsoc ) = max max Ssoc (x), max Ssoc (x) . (5.15) x∈[0,a−1]
x∈[a−1,a]
P 1 (x). Note that We first concentrate on maxx∈[0,a−1] Ssoc P1 P1 max Ssoc max Ssoc (x) = max (i − 1 + q) . x∈[0,a−1]
1≤i≤a−1 q∈[0,1]
(5.16)
P 1 (i − 1 + q) as a function of q. We fix an i ∈ {1, 2, . . . , a − 1} and we consider Ssoc Using elementary analysis we can easily see that there are three cases according to P (i) we have that the function the size of R. More concretely, for R tsoc P 1 Ssoc (i − 1 + q) is respectively decreasing, constant or increasing with respect to q. Hence ⎧ P1 P (i), S (i − 1), if R < tsoc ⎪ ⎪ ⎨ soc P1 P 1 (i − 1 + q), q ∈ [0, 1], if R = t P (i), max Ssoc (i − 1 + q) = Ssoc (5.17) soc ⎪ q∈[0,1] ⎪ ⎩ P1 P if R > tsoc (i). Ssoc (i), P 1 (x) in [0, a − 1] is taken in some In light of (5.17) we have that the maximum of Ssoc integer value of x so (5.16) becomes P1 P1 P1 P1 (x) = max{Ssoc (0), Ssoc (1), . . . , Ssoc (a − 1)}. max Ssoc
x∈[0,a−1]
(5.18)
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P 1 (x) only for integer values of x in [0, a − 1]. So we will now study the function Ssoc (5.13) assumes then the form
P1 Ssoc (i) =
Rμ i−1 )] ρ i+1 C( Rμ C − i) − ρC[ C − (1 + ρ + · · · + ρ , ρ i+1 − 1
i = 0, 1, . . . , a − 1.
(5.19) P 1 (i) is unimodal and so a local maximum We will prove now that the function Ssoc P 1 (i) − S P 1 (i − 1), we of it is also a global maximum. Studying the increments Ssoc soc obtain after some calculations that P1 P1 Ssoc (i) − Ssoc (i − 1) ≥ 0
⇔
P R ≥ tsoc (i),
(5.20)
P (i), i = 1, 2, . . . , a − 1 are given by (5.11). The sequence t P (i) for where tsoc soc i = 1, 2, . . . , a − 1 is easily seen to be increasing with respect to i which shows P 1 (i). Moreover, regarding the maximum i P 1 of S P 1 (i) in the unimodality of Ssoc soc soc {0, 1, . . . , a − 1} we have two cases: P (a − 1) then the maximum i P 1 is characterized by the relationship If R < tsoc soc P P1 P P1 tsoc (isoc ) ≤ R < tsoc (isoc + 1).
(5.21)
P (a − 1) then the maximum i P 1 is attained in a − 1. Hence we obtain If R ≥ tsoc soc ⎧ P1 P (i) ≤ R < t P (i + 1), i = 1, 2, . . . , a − 2, ⎨ Ssoc (i), if tsoc soc P1 max Ssoc (x) = ⎩ S P 1 (a − 1), if R ≥ t P (a − 1). x∈[0,a−1] soc soc (5.22) P 2 (x). Consider the right side of (5.14) as We now concentrate on maxx∈[a−1,a] Ssoc a function of q. Then we have
d P2 (1 − ρ)ρ a Caρ a Ssoc (a − 1 + q) = Rμ − . dq 1 − ρa (1 − qρ a )2
(5.23)
This derivative equals zero for qsoc given by the first branch of (5.12). a) P P2 If qsoc < 0 which happens if and only if R < aC(1−ρ μ(1−ρ) = tsoc (a), then Ssoc (a − P 2 (a − 1 + q) = 1 + q) is decreasing in [0, 1] and therefore we have that maxq∈[0,1] Ssoc P 2 P Ssoc (a − 1). Then by (5.15), (5.19), (5.22) and the continuity of Ssoc (x) (which gives P 2 (a − 1) = S P 1 (a − 1)) we have in particular Ssoc soc P P Ssoc (xsoc ) = max =
max
i∈{0,1,...,a−1}
P1 P2 Ssoc (i), Ssoc (a − 1)
P 1 (i), Ssoc
P (i) ≤ R < t P (i + 1), i = 1, 2, . . . , a − 2, if tsoc soc
P 1 (a − 1), Ssoc
P (a − 1). if R ≥ tsoc
(5.24)
This corresponds to the cases (a)–(b) of the statement of the theorem. Note that in the P (i) for some i = 1, 2, . . . , a − 1, the maximum is taken for all case (a) where R = tsoc x ∈ [i − 1, i] as we have seen in (5.17).
Queueing Syst (2008) 59: 237–269
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If qsoc ∈ [0, 1) which happens if and only if
aC(1−ρ a ) μ(1−ρ)
P (a) ≤ R < = tsoc
P 2 (a − 1 + q) has a unique maximum at q then Ssoc soc in [0, 1] and thereP 2 P 2 fore we have that maxq∈[0,1] Ssoc (a − 1 + q) = Ssoc (a − 1 + qsoc ). Then by (5.15), (5.19), (5.22) we have that P P P1 P2 Ssoc (xsoc ) = max max Ssoc (i), Ssoc (a − 1) aC μ(1−ρ)(1−ρ a ) ,
i∈{0,1,...,a−1}
P1 P2 (a − 1), Ssoc (a − 1 + qsoc )} = max{Ssoc P2 = Ssoc (a − 1 + qsoc ).
(5.25)
This corresponds to the case (c) of the theorem. aC P2 If qsoc ≥ 1 which happens if and only if R ≥ μ(1−ρ)(1−ρ a ) , then Ssoc (a − 1 + q) is P 2 (a − 1 + q) = S P 2 (a). increasing in [0, 1] and therefore we have that maxq∈[0,1] Ssoc soc As in the other cases we have P P P1 P2 Ssoc (xsoc ) = max max Ssoc (i), Ssoc (a) i∈{0,1,...,a−1}
P1 P2 P2 (a − 1), Ssoc (a)} = Ssoc (a). = max{Ssoc
(5.26)
This corresponds to the case (d) of the theorem. If ρ ≥ 1 then the case (d) never occurs because we have necessarily qsoc < 1. The case (c) becomes (c ). For the case ρ = 1 the reasoning is exactly the ksame but the formulas change a bit because the geometric sums of the type i−1 k=0 ρ are equal to i
i instead of 1−ρ 1−ρ . We can obtain the corresponding results by taking limits as ρ → 1 in the formulas for ρ = 1. Using the results of Theorems 5.2 and 5.3 we have the following figures that show the equilibrium and social optimal thresholds under the P information as R varies. P versus R. (a) Equilibrium pure and mixed strategies: Thresholds xind q 1 1+q • 2C μ
C μ
2
2+q •
2C μ
•
•
•
3C μ
q 1 1+q 2 2+q • • C μ
•
•
•
•
aC μ
a−1
a−1+qind
(a−1)C μ
where qind =
1−
Ca Rμ ρa
.
P a−1+qind xind
•
(a−1)C μ
•
3C μ
a−1
• aC μ
R(ρ≥1)
•
aC μ(1−ρ a )
a
P xind
R(ρ
