On optimal call admission control in cellular networks - Springer Link

Wireless Networks 3 (1997) 29–41

29

On optimal call admission control in cellular networks Ramachandran Ramjee a , Don Towsley a and Ramesh Nagarajan b a Department

of Computer Science, University of Massachusetts, Amherst, MA 01003, USA b Bell Labs, Lucent Technologies, Holmdel, NJ, USA

Two important Quality-of-Service (QoS) measures for current cellular networks are the fractions of new and handoff “calls” that are blocked due to unavailability of “channels” (radio and/or computing resources). Based on these QoS measures, we derive optimal admission control policies for three problems: minimizing a linear objective function of the new and handoff call blocking probabilities (MINOBJ), minimizing the new call blocking probability with a hard constraint on the handoff call blocking probability (MINBLOCK) and minimizing the number of channels with hard constraints on both of the blocking probabilities (MINC). We show that the well-known Guard Channel policy is optimal for the MINOBJ problem, while a new Fractional Guard Channel policy is optimal for the MINBLOCK and MINC problems. The Guard Channel policy reserves a set of channels for handoff calls while the Fractional Guard Channel policy effectively reserves a non-integral number of guard channels for handoff calls by rejecting new calls with some probability that depends on the current channel occupancy. It is also shown that the Fractional policy results in significant savings (20–50%) in the new call blocking probability for the MINBLOCK problem and provides some, though small, gains over the Guard Channel policy for the MINC problem. Further, we also develop computationally inexpensive algorithms for the determination of the parameters for the optimal policies.

1. Introduction During the last few years there has been tremendous interest and progress in the field of wireless communications. There is, hence, a great demand for Personal Communication Services (PCS) which will provide reliable voice and data communications, anytime and anywhere, via small lightweight, pocket-size terminals. PCS can be provided, in theory, by the current cellular technology and infrastructure and hence will be the focus of this paper. The service area in a PCS network is partitioned into cells1 . In each cell, a Base Station (BS) is allocated a certain number of channels2 which is assigned to the Mobile Subscriber (MS) enabling the MS to communicate with other MS’s or Public Switched Telephone Network (PSTN) users. Note that the BS itself is assigned a set of channels and this assignment could be static or dynamic. We primarily assume a static assignment of channels for this paper but the ideas in the paper can be extended easily to the dynamic assignment scenario as well. As the MS moves from one cell to another, any active call needs to be allocated a channel in the destination cell. This event, termed the handover or handoff, must be transparent to the MS. If the destination cell has no available channels, the on-going call is terminated. The disconnection in the middle of a call is highly undesirable and one of the goals of the network designer is to keep the probability of such an occurrence (also termed the handoff blocking probability) low. On the other hand, reserving channels for handoff traffic could increase blocking for new calls. As a result, there is a trade-off between the two Quality1 2

Both micro and macro cells are likely. Channels could be frequencies, time slots or codes depending on the radio technology used.

 J.C. Baltzer AG, Science Publishers

of-Service (QoS) measures, the handoff and the new call blocking probabilities. In this paper, we consider the optimal admission control policies for three problems based on these two QoS measures: MINOBJ: Minimizing a linear objective function of the two blocking probabilities. MINBLOCK: For a given number of channels, minimizing the new call blocking probability subject to a hard constraint on the handoff blocking probability. MINC: Minimizing the number of channels subject to hard constraints on the new and handoff call blocking probabilities. MINOBJ tries to minimize penalties associated with blocking new and handoff calls. Thus, MINOBJ appeals to the network provider since minimizing penalties results in maximizing the net revenue obtained. MINBLOCK places a hard constraint on handoff call blocking thereby guaranteeing a particular level of service to already admitted users while trying to maximize the net revenue. MINC is more of a network design problem where resources need to be allocated apriori based on, for example, traffic projections. The notion of guard channels was introduced in the mid80s [5,12] as a call admission mechanism to give priority to handoff calls over new calls. In this policy, a set of channels called the guard channels are permanently reserved for handoff calls. We show that this well-known Guard Channel policy is optimal for the MINOBJ problem. We then introduce the Fractional Guard Channel policy which effectively reserves a non-integral number of guard channels for handoff calls by rejecting new calls with some probability that depends on the current channel occupancy. We show that a restricted version of this Fractional policy

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R. Ramjee et al. / On optimal call admission control in cellular networks

is optimal for the MINBLOCK and MINC problems. Another important aspect of the optimal call admission policy question is the design of algorithms for determining the optimal settings for parameters associated with the policies. These algorithms could be employed either in off-line engineering exercises or for real-time admission control. In either of these cases and in particular for the latter, it is important that the algorithms be simple and efficient. To this end, we develop simple and computationally inexpensive algorithms for the determination of the optimal parameters of the Fractional policy. The remainder of the paper is organized as follows. In section 2, we describe the Guard Channel and the Fractional Guard Channel policies and compute blocking probabilities under these two policies. In section 3, we consider the problem of minimizing a linear objective function of the new and handoff call blocking probabilities and show that the Guard Channel policy is optimal for this problem. In section 4, we consider the MINBLOCK problem. We introduce a Limited Fractional Guard Channel policy, show that it is optimal, and develop an algorithm for the solution of the MINBLOCK problem. We show that the algorithm gives us significant improvement over the Guard Channel policy. In section 5, we consider the MINC problem and develop an algorithm for its solution. We present our conclusions in the final section and mention some further ideas that we are currently exploring.

2. Admission control policies and blocking performance We introduce the Guard Channel and Fractional Guard Channel admission control policies in this section. The computation of the blocking probability under these policies is detailed and will be useful in subsequent sections for proving the optimality of these policies. Consider a cellular network with C channels in a given cell. The Guard Channel policy reserves a subset of these channels (say C −T ) for handoff calls. Whenever the channel occupancy exceeds a certain threshold (T ), the Guard Channel policy rejects new calls until the channel occupancy goes below the threshold. In the Fractional Guard Channel policy, new calls are accepted with a certain probability that depends on the current channel occupancy. Thus, for each state, we have a randomization parameter which denotes the probability of acceptance of a new call. Note that both policies accept handoff calls as long as channels are available. These policies are illustrated algorithmically in figure 1. We next focus on the new and handoff call blocking probabilities under these policies. 2.1. Blocking performance We compute the performance of the admission policies based on the following assumptions:

Figure 1. Call admission policies.

• The arrival process of new and handoff calls is Poisson with rate λ1 and λ2 respectively. Let λ = λ1 + λ2 and λ2 = α ∗ λ. • The channel holding time for both type of calls is exponentially distributed with mean 1/µ; let ρ = λ/µ. • The busy-line effect [7] is negligible, i.e., the interval of time between two calls from a MS is much greater than the mean call holding time. This set of assumptions has been found to be reasonable as long as the number of mobile users in a cell is much greater than the number of channels and have been used in the models in [12,15,17]. We further assume that the cellular network is homogeneous and thus, we can examine a single network cell in isolation. Let us define the state of a cell at time t by the total number of occupied channels3. Thus, the cell channel occupancy can be modeled by a continuous time Markov chain with C states. The state transition rate diagram of a cell with C channels and C − T guard channels is shown 3

One could possibly enhance the state description by keeping track of new calls and handoff calls separately, rather than the total occupancy alone. However, this new state descriptor is not expected to change any of the conclusions of the paper given the memoryless nature of the arrival process.

R. Ramjee et al. / On optimal call admission control in cellular networks λ

λ

0

λ

1

αλ

2

µ

αλ

T

2µ

T+1

Τµ

αλ C-1

(T+1) µ

µ

(C-1)

3. Minimizing a linear objective function C

Cµ

Figure 2. State transition diagram (Guard Channel policy). γ λ 1 0

γ λ 2 1

µ

γ λ Τ

γ Τ+1 λ

2

T

2µ

Τµ

γ Τ+2 λ

T+1 (T+1) µ

γ λ C-1

T+2 (T+2) µ

γ λ C

C-1 (C-1)

µ

31

C Cµ

{γ i = α + ( 1 − α ) βi }

Figure 3. State transition diagram (Fractional Guard Channel policy).

in figure 2. Given this, it is straight forward to derive the steady-state probability Pj , that j channels are busy [11,12]:  j   ρ P0 , 0 6 j 6 T,  j! Pj = ρj αj−T    P0 , T 6 j 6 C, j!

In this section, we consider the problem of finding an admission control policy that minimizes a linear objective function of the new and handoff call blocking probabilities. Note that the assumptions in section 2.1 are applicable here. Consider any policy π that determines the acceptance or rejection of new and handoff calls. Let constants A1 and A2 denote the penalties associated with rejecting new and handoff calls respectively. Note that we are only interested in values of A1 and A2 such that 0 < A1 < A2 since we would like to give handoff calls higher priority than new calls. Let π1n (π2n ) be 0 or 1 depending on whether the nth new (handoff) call is accepted or rejected respectively. Then, we define ! N −1 N −1 X X 1 φπ = lim E A1 π1n + A2 π2n . (3) N →∞ N n=0

n=0

We are interested in determining optimal policy π ∗ over the set of all call admission control policies π. Problem MINOBJ. Find policy π ∗ such that φπ∗ = min φπ .

where

π

P0 = PT

ρj j=0 j!

+

1 PC

ρj αj−T j=T +1 j!

∗

.

The state transition rate diagram for the Fractional Guard Channel policy is shown in figure 3. At state i, a new call is accepted with a probability βi , 0 6 i 6 C, and handoff calls are accepted with probability 1. Thus the arrival rate at state i is (α+(1−α)∗βi )∗λ. The steady-state probabilities are calculated similarly as Q ρj ji=1 γi Pj = P0 , 0 6 j 6 C, j! where P0 = PC j=0

(ρj

1 Qj i=1

γi /j!)

and γi = α + (1 − α) βi ,

1 6 i 6 C.

Define β = (β1 , β2 , . . . , βC ) as the new call acceptance probability vector. Given the state probabilities, we can find the handoff blocking probability Bh (C, β) and new call blocking probability Bn (C, β) as Bh (C, β) = PC , C X Bn (C, β) = (1 − βj+1 )Pj ,

(1) where βC+1 = 0. (2)

j=0

Note that the blocking for the Guard Channel policy, with threshold T , can be calculated from the above by setting βi = 1, 1 6 i 6 T , and βi = 0, T + 1 6 i 6 C.

Note that policy π (if it exists) must exhibit performance that is better than or equal to that of the best Fractional Guard Channel policy. In [9], Miller obtains qualitative results regarding the optimal admission control policy that maximizes the total reward in a c-server queueing system with m customer classes, each class having a different reward associated with acceptance. Miller shows that if it is optimal to serve a customer of class 1 < k 6 m when i servers are free, then it is optimal to serve a customer of class k when j > i servers are free. For the case of a two class system with new and handoff calls, this result can be used to easily show that the Guard Channel policy is optimal for the MINOBJ problem. In this section, we present a different proof for the optimality of the Guard Channel policy for the MINOBJ problem. A trivial consequence of our approach is that the Guard Channel policy is shown to be optimal for a discounted cost finite and infinite horizon problem as well [2]. We first note that eq. (3) is a formulation for the average cost problem [2] with the cost of rejecting a handoff call being A2 and the corresponding cost for new call rejection being A1 . Let us consider the class of control policies that are stationary, i.e., depend only on the current state, which we denote as u(i). Given the stationary control policy, the state of the system now evolves as a Markov decision process [13] and in particular our interest will be only in stationary policies that lead to irreducible Markov chains. Before we proceed to the average cost problem, we first consider the below η-discounted finite horizon problem [2]: ! N −1 N −1 X X −ηTn1 −ηTn2 min E A1 e + A2 e , n=0

n=0

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R. Ramjee et al. / On optimal call admission control in cellular networks

where η is the discount factor and where Tn1 and Tn2 are the rejection times for the new and handoff calls respectively. We denote by Vkη (i) the optimal cost for the k-stage problem starting in state i (where i denotes the number of allocated channels). We now derive properties of this optimal cost function that will be useful for solving the average cost problem. We discretize the discounted return problem above by using the standard uniformization technique [2] and by scaling time, one can assume that Cµ + λ1 + λ2 + η = 14 . The optimal cost function V (·) then satisfies
η η Vkη (i) = λ1 min Vk−1 (i + 1), A1 + Vk−1 (i)
η η + λ2 min Vk−1 (i + 1), A2 + Vk−1 (i) η η + i µ Vk−1 (i − 1) + (C − i)µVk−1 (i), 0 6 i 6 C. (4) The first term is the contribution to the cost if the next transition is the arrival of a new call. Here, we have the option of rejecting the arrival in which case a cost A1 is incurred. The second term is the contribution to the cost for a handoff call arrival. Here, we have the option of rejecting the handoff call in which case a cost A2 is incurred. The third term is the contribution to the cost due to call completions and the fourth term is a consequence of the uniformization. The boundary conditions are handled by defining Vkη (C+ 1) = ∞ and Vkη (−1) = Vkη (0). Note that we are assuming control over both the new and handoff calls here. We will show later that controlling handoff calls is not beneficial as it leads to idling of some channels. We now state a lemma on some properties of the optimal cost function. The proof of the lemma can be found in appendix A. Lemma 1. Vkη (i) is monotonically non-decreasing and convex in i for all k. Given the lemma for the finite horizon optimal cost function, under minor technical conditions, it can be shown that the optimal cost function for the infinite-horizon problem, V η (i) = limk→∞ Vkη (i), is also non-decreasing and convex [16, assumption 8.28]. Also, since the state space is finite and the Markov chain induced by the control policy is irreducible, V η (i) − V η (0) is bounded uniformly [13]. We now return to consider the original average cost problem. Since V η (i) − V η (0) is bounded, the optimal control policy for the average cost problem is also stationary and the optimality equation is given as [13]:
g + h(i) = λ1 min h(i + 1), A1 + h(i)
+ λ2 min h(i + 1), A2 + h(i) + iµh(i − 1) + (C − i)µh(i), 0 6 i 6 C, (5) 4

Transitions associated with the discount cost η can be thought of as terminating the process of channel occupancy evolution.

where g = limη→1 (1 − η)V η (0) is a constant and h(i) = limr→∞ [V ηr (i)−V ηr (0)] for some sequence ηr → 1. Note that h(i) inherits the structural properties of V η (·) and hence is non-decreasing and convex. The optimal average cost policy is one which minimizes the right-hand side of the above equation. The following theorem shows that a threshold policy, the Guard Channel policy, is optimal for the average cost problem and hence for the MINOBJ problem. Theorem 1. Among all call admission control policies π, an optimal policy for the MINOBJ problem is of the threshold type (i.e., a Guard Channel policy). Proof. The optimal control policy u(i) that incurs the minimum cost (see eq. (5)) is given as  1, if h(i + 1) − h(i) 6 A1 , u(i) = 0, otherwise, where u(i) = 1(0) corresponds to accepting (rejecting) a new call when i channels in the cell are busy. Since h(i) is non-decreasing and convex, h(i) − h(i − 1) is non-decreasing in i, and hence there exist integers i0 and i1 where i0 = arg min{i: (h(i) − h(i − 1)) > A1 } and i1 = arg min{i: (h(i) − h(i − 1)) > A2 }. Thus, for new calls we have u(i) = 1 for i < i0 and u(i) = 0 for i > i0 and hence the optimal control policy for new calls is of the threshold type. Let us now consider the policy for handoff calls. Since A2 > A1 , i1 > i0 . If i1 < C, then this will result in idling one or more channels. We can, hence, do better by not blocking the handoff traffic and letting them use these channels, i.e., i1 = C. Thus the overall optimal policy is to block all new calls when the channel occupancy reaches or exceeds i0 and not block any handoff calls. In other words, the optimal policy is to reserve C −i0 channels for handoff calls. Thus, the optimal policy for the average cost and MINOBJ problem is the Guard Channel policy.  In the following sections, we consider the MINBLOCK and MINC problems. For these problems, we restrict our attention to a Limited Fractional Guard Channel policy and show that this policy is optimal.

4. Minimizing new call blocking with a hard constraint Problem MINBLOCK. Given C, minimize Bn (·) such that Bh (·) 6 Ph .

(6)

We consider a restricted version of the Fractional Guard Channel policy and show that it is the optimal policy for the MINBLOCK problem. 4.1. Limited Fractional Guard Channel policy Figure 4 shows the state transition rate diagram of a system with C channels for the Limited Fractional Guard

R. Ramjee et al. / On optimal call admission control in cellular networks λ 0

λ 1

µ

λ 2

2µ

(α+(1−α)β)λ T

Τµ

T+1 (T+1) µ

αλ

αλ

T+2 (T+2) µ

αλ C-1

(C-1)

µ

C Cµ

Figure 4. State transition diagram (Limited Fractional Guard Channel policy).

33

LFG policy, is optimal for the MINBLOCK problem. Let us first define some notation:  i j   X ρ , 0 6 i < C, Ui = j!   j=0 0, otherwise,  C−1 j   X ρ αj−i , 0 6 i < C, Vi = j!   j=i 0, otherwise,   ρC C−i α , 0 6 i 6 C, Wi =  C! 0, otherwise, and let Xi = Ui−1 + Vi + Wi . Consider the MINOBJ problem with A1 = 1. Define ET = Bn (·) + A2 Bh (·) for policy gT . In the following derivations, we are using results from the previous section that policy gT is optimal and characterizing the relationship between A2 and the threshold T . ET can be rewritten as ET =

Figure 5. Call admission with Limited Fractional Guard Channel policy.

Channel (LFG) policy and figure 5 outlines this call admission algorithm. As the name suggests, the LFG policy is a simplification of the more general Fractional Guard Channel policy described earlier. In the LFG policy, when the system is in state T , new calls are accepted with a probability β. From states T + 1 to C, handoff calls are accepted and in states 0 to T − 1, both types of calls are accepted. Thus, the randomization in the LFG policy is restricted to just one state as compared to the Fractional Guard Channel policy where randomization could potentially occur at every state. The handoff and new call blocking probabilities for the LFG policy can be easily calculated using eqs. (1) and (2) by setting βT +1 = β and the values of βi = 1, 1 6 i 6 T , and βi = 0, T + 1 < i 6 C. We now show that the LFG policy is optimal for the MINBLOCK problem.

T = 0, 1, . . . , C.

We first show that gT is optimal for MINOBJ if AT2 +1 6 A2 6 AT2 where  VT −1 XT − VT XT −1   − 1, T = 1, . . . , C, W X T T −1 − WT −1 XT AT2 = (7) 0, T = C + 1,    ∞, T = 0. We will first need the following property of the function AT2 . Property 1. AT2 > AT2 +1 for T = 0, 1, . . . , C. Proof. In appendix B.



Lemma 2. If AT2 +1 6 A2 6 AT2 , then policy gT is optimal for the MINOBJ problem. Proof. A2 > AT2 +1 VT XT +1 − VT +1 XT = −1 WT +1 XT − WT XT +1 VT +1 + WT +1 (A2 + 1) ⇒ XT +1 VT + WT (A2 + 1) > XT ⇒ ET +1 > ET

4.2. Optimality of the LFG policy The proof methodology that we adopt here is similar to the one Nain and Ross introduced to show the optimality of the randomized µC rule in the presence of a hard constraint [10]. We begin with the MINOBJ problem and derive some properties on the threshold parameter for the Guard Channel policy. We then apply the constraint and show that the Limited Fractional policy is optimal. Let gi represent the Guard Channel policy with threshold i. Thus, we have a set of policies g0 , g1 , . . . , gC for a C channel system. We know that, for the MINOBJ problem, ∃i such that gi is optimal. We show below that a policy that randomizes between two static policies gi−1 and gi , viz. an

VT + WT (A2 + 1) , XT

and A2 6 AT2 VT −1 XT − VT XT −1 = −1 WT XT −1 − WT −1 XT

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R. Ramjee et al. / On optimal call admission control in cellular networks

VT −1 + WT −1 (A2 + 1) XT −1 VT + WT (A2 + 1) > XT ⇒ ET −1 > ET . ⇒

Also, if ET 6 ET +1 , then ET 6 ET +i , i = 1, . . . , C − T , since ET +i−1 6 ET +i follows from property 1. Similarly, if ET 6 ET −1 , then ET 6 ET −i , i = 1, . . . , T , since ET −i+1 6 ET −i follows from property 1. Thus, ET is minimum among all Ei , i = 0, 1, . . . , C, and policy gT is optimal for problem MINOBJ.  Consider the Limited Fractional Guard Channel policy fk,q := [gk−1 , gk , q],

q ∈ [0, 1],

which admits new calls at state k − 1 with probability q, and admits only handoff calls from states k through C. Let Bh (u) and Bn (u) be the handoff and new call blocking probabilities for a stationary (possibly randomized) policy u. Observe that, Bh (fk,0 ) = Bh (gk−1 ), Bh (fk,1 ) = Bh (gk ) and Bh (fk,q ) is a continuous function of q over interval [0,1] for k = 0, 1, . . . , C. We now state a lemma which will be useful in the subsequent theorem that proves that fk,q is optimal for the MINBLOCK problem.

must be at most Ph , i.e., it determines the optimal values of parameters T and β for a given constraint Ph . We first establish some properties of Bn (·) and Bh (·) which will be useful in proving that algorithm MINB minimizes the new call blocking probability while satisfying the constraint on the handoff call blocking probability. Lemma 4. Bn (·) (Bh (·)) is a monotonically decreasing (increasing) function of both T and β and hence also of T +β. Proof. Oh and Tcha [11] have shown that Bh (C, T + 1, β) > Bh (C, T , β) and Bn (C, T + 1, β) < Bn (C, T , β) for the Guard Channel policy. Thus, we only need to show that for a given T , Bn (Bh ) decreases (increases) monotonically with β. Differentiating Bh with respect to β, ∂Bh (·) = ∂β

PT







ρC ρj j=0 j! ∗ C! ∗ (1 − α)
PT ρj PC γρj αj−T −1 2 j=0 j! + j=T +1 j!

> 0. (8)

Since the derivative of Bh with respect to β is postive, Bh is a monotonically increasing function of β. Similarly, differentiating Bn with respect to β, we can show that the derivative is negative. Thus, we can prove that Bn decreases monotonically with β. 

Lemma 3. Consider the objective function J ω (u) = Bn (u) + ωBh (u) under admission control policy u. If ω = AT2 , where AT2 is given by eq. 7, policies gT −1 , gT and fT ,q are all optimal in minimizing the objective function.

Theorem 3. Algorithm MINB minimizes the new call blocking probability (Bn (·)) while satisfying the constraint on handoff call blocking probability (Bh (·) 6 Ph ).

Proof. If AT2 6 ω 6 AT2 −1 , policy gT −1 is optimal by the previous lemma. Thus, if ω = AT2 , policies gT −1 and gT are both optimal. Hence fT ,q , which randomizes between two optimal policies, is also optimal. 

Proof. Clearly, if the condition in step 3 of algorithm MINB (see figure 6) is true, we are done; we don’t have

Theorem 2. If Bh (g0 ) 6 Ph 6 Bh (gC ), then for some q ∈ [0, 1], j = min{i: Bh (gi ) 6 Ph }, fj,q is optimal over all call admission control policies for the MINBLOCK problem. Proof. Suppose Bh (gj−1 ) 6 Ph 6 Bh (gj ) with j as given above. By continuity of fj,q , ∃q ∈ [0, 1] such that Bh (fj,q ) = Ph . Moreover, fj,q minimizes J γ (u) where γ = Aj2 by lemma 3. Thus, for any policy u, Bn (fj,q ) + γPh = J γ (fq ) 6 J γ (u) 6 Bn (u) + γBh (u). Thus Bn (fq ) − Bn (u) 6 γ(Bh (u) − Ph ) which implies Bn (fq ) 6 Bn (u) for any policy which is feasible (i.e., Bh (u) 6 Ph ). Thus fj,q , a Fractional policy with parameters j, q is optimal for the MINBLOCK problem.  4.3. Algorithm for optimal parameters of LFG policy We now present an algorithm (figure 6), labeled MINB, that minimizes the new call blocking probability for LFG with the constraint that the handoff call blocking probability

Figure 6. Algorithm MINB.

R. Ramjee et al. / On optimal call admission control in cellular networks Table 1 MINBLOCK: minimize Bn such that Bh 6 0.01.

New and Handoff Call Blocking Probabilities 1.00

Bn (12,T,beta) B (12,T,beta) h

0.90

Item Case Case Case Case Case

0.80

0.70

Probability

35

Arrival Handoff C Guard policy Fractional policy Gain (Bn ) (%) rate (λ) prob (α) (Bn ) I 6 II 7 III 12 IV 14 V 14

1/6 2/7 1/6 2/7 2/7

12 13 19 22 23

0.024859 0.031118 0.029255 0.023117 0.007445

0.013313 0.021538 0.022536 0.015211 0.007445

46.45 30.78 22.97 34.20 0.0

0.60

0.50

0.40

0.30

0.20

0.10

0.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

T+Beta

Figure 7. Bh and Bn as functions of T + β.

any guard channels in this case and this would result in the smallest blocking probability for new calls. Proceeding beyond step 3, the algorithm uses the fact that finding the value of T + β which satisfies the equation Bh (·) = Ph is precisely the value for which Bn (·) is minimized. This is based on the fact that Bh (·) is a monotonically increasing function of T + β and hence the value of T + β which satisfies the equality on the handoff constraint is the largest admissible value for T + β. This also minimizes the new call blocking probability as Bn (·) is monotonically decreasing function of T + β. The while loop in step 4 performs a binary search to locate the value of T + β which satisfies the equation Bh (·) = Ph .  A graph of the functions Bh and Bn versus T + β is shown in figure 7 for illustration. The traffic parameters correspond to those of case I in table 1 with 12 channels. Finally, the computational complexity of the algorithm is simply O(log2 (C/ε)) where ε is the desired resolution for the binary search. We have assumed that each iteration of the while loop consumes one unit of work (this involves 3 comparisons and 2 assignments). 4.4. Numerical examples We show below that the introduction of Fractional Guard Channels gives us significant savings in terms of new call blocking probability as compared to the Guard Channel approach. The new and handoff call arrival rates (normalized by the call holding times), and the number of channels used in table 1 are taken from [11] where the efficiency of the well-known Guard Channel policy is examined. These parameters can be determined at the base-station by hav-

ing a running estimate of the new and handoff call arrival rates using standard techniques such as exponential smoothing [3]. The handoff call arrival rates can also be deduced based on an expected new call arrival rate and some of the direct cellular network parameters that can be measured including the cell-size, and the speed and the average cell resident time of the mobile user. If we assume that the cell duration time is exponentially distributed, we can derive the call handoff rates by solving a set of equations by repeated substitution [8]. Our main interest in this paper is to examine the efficiency of call admission control schemes and thus, we illustrate the efficacy of our algorithms on some of the new and handoff call arrival rates used in the literature. In table 1, column 5 is obtained by computing the minimum new call blocking probability Bn for the Guard Channel policy subject to the constraint Bh 6 0.01. Column 6 of table 1 is obtained by calculating the new call blocking probability for the policy given by algorithm MINB. The last column in the table lists the percentage gain in new call blocking for LFG over the Guard Channel policy. As can be seen from cases I through IV in table 1, significant reductions in the new call blocking probability can be achieved by adopting the LFG policy instead of the Guard Channel policy. The values in case V have been particularly chosen to illustrate a case when the constraint on handoff probability can be met without the use of any guard channels. In this case, as expected, there is no gain for the LFG policy. We also did some limited experiments to verify the time-complexity of algorithm MINB. We consider case IV in table 2. The computation of the minimum value of Bn (·) in this case took 0.8 milliseconds on a Sparc 10 workstation. Thus, the algorithm is extremely fast and can be used in the base-station for real-time admission control. 5. Minimizing the number of channels with hard constraints Lastly, we consider the problem of finding an admission control policy that minimizes the number of channels while satisfying the blocking constraints for both the new and handoff calls. Formally, Problem MINC. Minimize C such that Bn (C, T , β) 6 Pn

(9)

Bh (C, T , β) 6 Ph ,

(10)

and

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R. Ramjee et al. / On optimal call admission control in cellular networks

with 0 6 T 6 C (T , C ∈ I) and 0 6 β < 1. Note that the LFG policy is optimal for this problem also. This follows easily from the fact the LFG policy is optimal for MINBLOCK problem and MINC problem can be reformulated as find the minimum C such that the minimum Bn (from the MINBLOCK problem) is smaller than Pn , given Bh 6 Ph . 5.1. Algorithm for optimal parameters of LFG policy We now present an algorithm (figure 8) that calculates the minimum number of channels which satisfy the QOS constraints that the new call blocking probability must be at most Pn and handoff call blocking probability must be at most Ph . Note that we could use algorithm MINB by starting with the value of C that meets the constraint for the handoff call blocking probability and increase C until the new call blocking probability also meets its constraint. Algorithm MIN detailed in figure 8 is more efficient. The following claims will be useful in proving that algorithm MIN finds the minimum number of channels that satisfies the constraints expressed in eqs. (9) and (10).

Claim 1. There exists a value of T +β for C > Clb , where Clb = arg min{C: Bn (C, C, 0) 6 Pn }, such that Bn (·) and Bh (·) either both violate their constraints, Pn and Ph , or both meet them. Let Tn + βn be such that Bn (C, Tn , βn ) = Pn . Note that a feasible solution Tn + βn always exists in [0, C] for C > Clb since Bn (C, 0, 0) = 1.0 and Bn (C, C, 0) 6 Pn and Bn (·) is monotonically decreasing in T + β. Now consider Bh (·). If Bh (C, T , β) > Ph , ∀(T + β) ∈ [0, C], then both Bn (·) and Bh (·) violate their constraints in the interval [0, Tn + βn ). On the other hand, if Bh (C, T , β) 6 Ph , ∀(T + β) ∈ [0, C], then both Bn (·) and Bh (·) meet their constraints in the interval [Tn + βn , C]. Finally, if there is a feasible solution to Bh (C, Th , βh ) = Ph for Th + βh ∈ [0, C], then if Th + βh > Tn + βn , the interval [Tn +βn , Th +βh ] includes all points T +β where both Bh (·) and Bn (·) meet their constraints. If Th + βh 6 Tn + βn , then both Bh (·) and Bn (·) will violate their constraints for T + β ∈ [Th + βh , Tn + βn ]. Claim 2. If there exists a value of T + β for a given C where both the constraints given by eqs. (9) and (10) are violated, then there is no value of T + β for the given C where both the constraints can be met. Claim 2 can be inferred from the arguments for different cases in proof of claim 1. A simpler argument is that if a value of T + β violates both the constraints, reducing (increasing) the value will lead to larger violations of Bn (·) (Bh (·)). Theorem 4. Algorithm MIN finds the minimum number of channels that satisfies the constraints given by eqs. (9) and (10).

Figure 8. Algorithm MIN.

Proof. The initial assignment in step 2 of the algorithm is a lower bound, Clb , on the minimum number of channels. If the condition at step 3 succeeds, we are done. If not, the while loop in step 6 tries to locate a value for T and β (i in algorithm MIN represents T + β) which satisfies the given constraints. Since we consider only values of C > Clb , claim 1 holds and hence the algorithm is guaranteed to terminate provided the algorithm can find the point that either violates or meets the constraints. In the following, we show that the algorithm finds such a point. To do that, we show that at each iteration of the while loop, the following invariant holds. Invariant: The interval represented by [L, U ] is such that for T + β = L, the constraint given by eq. (10) is met and for T + β = U , the constraint given by eq. (9) is met. This is due to the fact that the iteration condition of the while loop ensures that exactly one of the constraints is met at the midpoint of the interval and the body of the while loop resets the endpoint where that condition had been met previously to the midpoint. For example, if the constraint given by eq. (9) is violated at the midpoint while

R. Ramjee et al. / On optimal call admission control in cellular networks

37

Table 2 Minimize C such that Bn 6 0.02 and Bh 6 0.01. Item Case Case Case Case

I II III IV

Arrival rate (λ)

Handoff prob (α)

Guard policy (C, T , Bn , Bh )

Fractional policy (C, T + β, Bn , Bh )

Gain (∆C)

6 7 12 14

1/6 2/7 1/6 2/7

(13,12,0.012,0.001) (14,13,0.016,0.002) (20,19,0.018,0.002) (23,22,0.015,0.002)

(12,11.63,0.016,0.008) (14,14.0,0.007,0.007) (20,20.0,0.01,0.01) (22,21.31,0.02,0.006)

1 0 0 1

C ∗ is the optimal number of channels for problem MINC under the Fractional Guard Channel policy, and Th + βh (Tn + βn ) is the root of the equation Bh (Th + βh ) = Ph , (Bn (Th + βn ) = Pn ), and ε = |Th + βh − Tn − βn | then the worst case complexity of the algorithm is simply O(C ∗ log2 (C ∗ /ε)). The C ∗ factor arises as we successively iterate through various C values and the log(·) factor is due to the while loop for each value of C. 5.2. Numerical examples

Figure 9. The region [11.37–11.84] where Bh 6 0.01 and Bn 6 0.02.

the constraint given by eq. (10) is satisfied, the lower bound of the interval is moved to the midpoint resulting in a new interval where the invariant still holds. Thus, an interval [L1 , U1 ] is halved at each iteration, to form a new interval [L2 , U2 ], while the invariant is maintained. Since the invariant is maintained, it is easy to see that a point that either meets the constraints or violates them both cannot be in the part of the interval, [L1 , U1 ]−[L2 , U2 ], that is eliminated from future consideration due to the monotonicity of Bh (·) and Bn (·). Hence, the algorithm is guaranteed to find the point (which is guaranteed to exist by claim 1) that either meets the constraints or violates it. In the latter case, by claim 2, there is no feasible solution for C and hence we increment C by 1 and go to step 3. Thus, algorithm MIN finds the minimum number of channels that satisfies the required QOS constraints.  A graph of the functions Bh and Bn versus T + β is shown for illustrative purposes in figure 9. The traffic parameters correspond to those of case I in table 2 with 12 channels. The figure highlights the region of figure 7 in which the QoS constraints are met. Finally, we consider the computational complexity of algorithm MIN. If

We show below that the introduction of Fractional Guard Channels yields smaller values for the number of channels compared to the Guard Channel policy. The load values in table 2 are taken from [11]. Column 4 is obtained via an algorithm detailed in [11] which evaluates the minimum value of C for the Guard Channel policy while meeting the blocking constraints. Column 5 is obtained via algorithm MIN. From table 2, we can see that the LFG policy yields a smaller value of C in case I (12 instead of 13) and case IV (22 instead of 23). However, the LFG policy in general appears to provide only small improvements over the Guard Channel policy. Further examination revealed that an increase in capacity of a single channel results in significant reductions in blocking probabilities since we are dealing with fairly small values of blocking probabilities. Thus, the small increase in number of channels reflects significant gains in reduction of blocking probabilities. Finally, in order to gauge the time complexity of MIN, we consider case IV in table 2. In this case, the optimal C ∗ was computed in about 0.6 milliseconds on a Sparc 10 workstation. Hence, the algorithm is indeed extremely fast.

6. Conclusions and future work In this paper, we considered optimal admission control policies in cellular networks in the light of three problems: minimizing a linear objective function of the new and handoff call blocking probabilities (MINOBJ), minimizing the new call blocking probability with a hard constraint on handoff call blocking probability (MINBLOCK) and minimizing the number of channels with hard constraints on both the blocking probabilities (MINC). We showed that the well-known Guard Channel policy was optimal for MINOBJ. We defined a new Fractional Guard Channel policy and showed that a restricted version of it was optimal for the MINBLOCK and MINC problems. We also showed that

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R. Ramjee et al. / On optimal call admission control in cellular networks

the Fractional Guard Channel policy resulted in significant (20–50%) savings in the new call blocking probability over the Guard Channel policy for the MINBLOCK problem and provided some, though small, improvement over the Guard Channel policy for the MINC problem. Further, we showed that the algorithms developed in this paper for the solution of these problems were computationally inexpensive and have potential for use in engineering exercises as well as in real-time control mechanisms for cellular networks. We are exploring the use of the Fractional Guard Channel policy to assist in Dynamic channel assignment and other hybrid policies [14]. One can also easily extend this policy to have finite buffers for handoff and/or new calls [6] in case delaying handoff and/or new calls is considered acceptable. We are also studying call admission policies when there are multiple classes of wireless traffic. Finally, we are exploring various issues related to employing the Fractional Guard Channel policy in a real-time control scenario.

Acknowledgements We would like to thank Professor Jim Kurose and Dr. Thomas La Porta for the many discussions and suggestions which served to influence and improve this work.

and fourth terms of ∆(i) (we have factored out the common term µ from the following equations). η η (C − i)Vk−1 (i) + i Vk−1 (i − 1) η η (i − 2) − (C − (i − 1))Vk−1 (i − 1) − (i − 1)Vk−1 η η = (C − i)Vk−1 (i) − (C − i)Vk−1 (i − 1) η η + (i + 1)Vk−1 (i − 1) − (i − 1)Vk−1 (i − 2) η η > (i + 1)Vk−1 (i − 1) − (i − 1)Vk−1 (i − 2) η (since Vk−1 (i) is non-decreasing) η η > i Vk−1 (i − 1) − i Vk−1 (i − 2) > 0.

Hence, ∆(i) = Vkη (i) − Vkη (i − 1) > 0 and thus Vkη (i) is a monotonically non-decreasing function of i for all k. Next, we show Vkη (i) is convex for i = 0, . . . , C. This is equivalent to showing ∆(i) < ∆(i + 1) for i = 1, . . . , C − 1 which we show by induction. The basis step is again trivial since V0η (i) = 0, 1 6 i 6 C and V0η (C + 1) = ∞. η Assume Vk−1 (i) is convex. We need to show that Vkη (i) is convex. The first and second terms in eq. (4) can be shown to be convex by induction (8.35 in [16]). The arguments for the 3rd and 4th terms follow (we have again η factored the common term µ). Let Vk−1 (i) be convex for i = 1, . . . , C − 1. Let δ(i) represent the 3rd and 4th terms of Vkη (i) − Vkη (i − 1). Then η η δ(i) = i Vk−1 (i − 1) + (C − i)Vk−1 (i) η − (i − 1)Vk−1 (i − 2) η − (C − (i − 1))Vk−1 (i − 1)

Appendix A Proof of lemma 1. We first show that Vkη is monotonically non-decreasing by induction. The basis step is trivial since η V0η (i) = 0. Assume that Vk−1 (i) is monotonically nondecreasing. We need to show that Vkη (i) is monotonically non-decreasing. We are restating eq. (4) here for clarity:
η η Vkη (i) = λ1 min Vk−1 (i + 1), A1 + Vk−1 (i)
η η (i + 1), A2 + Vk−1 (i) + λ2 min Vk−1 η η + i µ Vk−1 (i − 1) + (C − i) µ Vk−1 (i), 0 6 i 6 C. The first term in ∆(i) = Vkη (i) − Vkη (i − 1) is (factoring out the common term λ1 ),
η η min Vk−1 (i + 1), A1 + Vk−1 (i)
η η − min Vk−1 (i), A1 + Vk−1 (i − 1)
η η η = Vk−1 (i) + min Vk−1 (i + 1) − Vk−1 (i), A1
η η η − Vk−1 (i − 1) − min Vk−1 (i) − Vk−1 (i − 1), A1 η η > Vk−1 (i) − Vk−1 (i − 1)
η η − min Vk−1 (i) − Vk−1 (i − 1), A1 η (since Vk−1 (i) is monotonically non-decreasing) η > 0 (since Vk−1 (i) is monotonically non-decreasing). Similarly, the second term of ∆(i) can also be shown to be monotonically non-decreasing. Now, consider the third

η η = (C − i)Vk−1 (i) − (i − 1)Vk−1 (i − 2) η + (2i + 1 − C)Vk−1 (i − 1).

Now, δ(i + 1) − δ(i) η η = (C − i − 1)Vk−1 (i + 1) − i Vk−1 (i − 1) η + (2i + 3 − C)Vk−1 (i) η η − (C − i)Vk−1 (i) + (i − 1)Vk−1 (i − 2) η (i − 1) − (2i + 1 − C)Vk−1 η η = (C − i − 1)Vk−1 (i + 1) + (3i + 3 − 2C)Vk−1 (i) η η − (3i + 1 − C)Vk−1 (i − 1) + (i − 1)Vk−1 (i − 2) η η = (C − i − 1)Vk−1 (i + 1) − 2(C − i − 1)Vk−1 (i) η + (C − i − 1)Vk−1 (i − 1) η η + (i + 1)Vk−1 (i) − 2i Vk−1 (i − 1) η + (i − 1)Vk−1 (i − 2) η η > (i + 1)Vk−1 (i) − 2i Vk−1 (i − 1) η η + (i − 1)Vk−1 (i − 2) (by convexity of Vk−1 (i)) η η > i Vk−1 (i) − 2i Vk−1 (i − 1) η η + i Vk−1 (i − 2) (since Vk−1 (i) is non-decreasing) η > 0 (by convexity of Vk−1 (i)). Thus, Vkη (i) is a convex function in i for all k.



R. Ramjee et al. / On optimal call admission control in cellular networks

Case 3. T = 1, 2, . . . , C − 1. Assume AT2 > AT2 +1 .

Appendix B Proof of property 1

⇒

Case 1. T = C. C+1 Assume AC = 0. 2 > A2 AC 2 >0

>

C−1 X j=0

VT −1 XT − VT XT −1 −1 WT XT −1 − WT −1 XT VT XT +1 − VT +1 XT > −1 WT +1 XT − WT XT +1

⇒ XT2 VT −1 WT +1 − XT XT +1 VT −1 WT − XT −1 XT VT WT +1 + XT −1 XT +1 VT WT | {z }

VC−1 XC − 0XC−1 ⇒ −1>0 WC XC−1 − WC−1 XC ⇒ XC (VC−1 + WC−1 ) > WC XC−1  C−1  C X ρj ρ ρC ⇒ + α j! (C − 1)! C! j=0

> XT2 VT +1 WT −1 − XT XT +1 VT WT −1 − XT −1 XT VT +1 WT + XT −1 XT +1 VT WT | {z }

ρj ρC ρC ρC + α j! C! C! C!

! C−1 X ρj ρC ρj ρC−1 ⇒ − j! (C − 1)! j! C! j=0 j=0 ! C X ρC ρC ρj ρC + α− α > 0. j! C! C! C! j=0 C X

The second term in the above equation is clearly greater than 0. The first term can further be expanded into  C−1   ρ ρC ρC + − (C − 1)! 1!(C − 1)! C!   C+1 ρ ρC+1 + − + ··· 2!(C − 1)! 1!C!   ρ2C−1 ρ2C−1 + − C!(C − 1)! (C − 1)!C!

⇒ XT2 (VT −1 WT +1 − VT +1 WT −1 ) > XT XT +1 (VT −1 WT − VT WT −1 ) + XT XT +1 (VT WT +1 − VT +1 WT ). Now, VT WT +1 − VT +1 WT

Case 2. T = 0. We need to show that A12 < A02 = ∞. It suffices to show that the denominator of A12 is not zero since the numerator is a known finite quantity. The denominator is W1 X0 − W0 X1

j=T

ρC C−1 = α C!

C X ρj j=0

j!

αj

C X

ρC C ρj j−1 α 1+ α C! j! j=1

j=T +1

and VT −1 WT +1 − VT +1 WT −1

j=T −1

ρ C−1 α (1 − α) C! > 0. =

Thus, the property holds for T = 0.

ρj j−T +1 ρC C−T −1 α α j! C!

C−1 X

ρj j−T −1 ρC C−T +1 α α j! C! j=T +1  T −1  ρC C−T −1 ρ ρT = α + α . C! (T − 1)! T ! −

Substituting, ρj X ρj j−T + α j! j! C



j=T

ρ ρT −1 × α+ T! (T − 1)! !

T

!

ρC C−T −1 α C!



! C X ρj j−T −1 ρC C−T ρT −1 + α α j! j! C! (T − 1)!

T X ρj

>

j=0

j=1

C

C−1 X

=

!

X ρj ρC C−1 ρj α 1+ αj − α − αj C! j! j!

ρj j−T −1 ρC C−T α α j! C!

ρC C−T −1 ρT = α C! T!

j=0

C

=

ρj j−T ρC C−T −1 α α j! C!

C−1 X

−

T −1 X

!

C−1 X

=

which is also greater than zero. Thus, the property holds for T = C.

−

39

+

T −2 X j=0

j=T +1

! C X ρj ρj j−T +1 ρC C−T −1 ρT + α α . j! j! C! T! j=T −1

Rearranging and collecting ρT /T ! and ρT −1 /(T − 1)! separately,

40

R. Ramjee et al. / On optimal call admission control in cellular networks

ρT ρC C−T −1 α T ! C! T −2 X

T −1 X j=0

C X

! C ρj X ρj j−T α + α j! j! j=T !)

ρj j−T +1 α j! j=0 j=T −1 ( T −1 ! C X ρj X ρT −1 ρC C−T −1 ρj j−T + α + α (T − 1)! C! j! j! j=0 j=T !) T C X X ρj ρj j−T −1 −α + α >0 j! j! j=0 j=T +1 ( T −1 C X ρT ρC C−T −1 X ρj ρj j−T +1 ⇒ α α+ α T ! C! j! j! j=0 j=T | {z } ) T −1 C X ρj X ρj − − αj−T +1 j! j! j=0 j=T | {z } ( T C X ρj j−T ρT −1 ρC C−T −1 X ρj + α + α (T − 1)! C! j! j! j=0 j=T +1 | {z } ) T C X ρj X ρj − α− αj−T > 0 j! j! j=0 j=T +1 | {z } −

j

(

ρ + j!

ρT −1 ρC C−T −1 α (1 − α) (T − 1)! C! ) ( T T −1 X ρj ρ X ρj × − >0 j! T j! j=0 j=0    ρT −1 ρC C−T −1 ρ ⇒ (1 − α) 1 + ρ − α (T − 1)! C! T  2   3  2 3 ρ ρ ρ ρ + − + − + ··· 2! T 3! T 2!   T −1 ρT −1 ρ + − > 0. (T − 1)! T (T − 2)!

⇒

The last equation is clearly true. Thus, property 1 holds for T = 0, 1, . . . , C. 

References [1] A. S. Acampora and M. Naghshineh, Control and quality-of-service provisioning in high-speed microcellular networks, IEEE Personal Communications 1(2) (1994) 36–43. [2] D. P. Bertsekas, Dynamic Programming: Deterministic and Stochastic Models (Prentice-Hall, Englewood Cliffs, NJ, 1987). [3] C. Chatfield, The Analysis of Time Series: An Introduction (Chapman-Hall, New York, 3rd ed., 1984). [4] D. C. Cox, Personal communications – a viewpoint, IEEE Communications Magazine 28(11) (1990). [5] D. Hong and S. S. Rappaport, Traffic model and performance analysis for cellular mobile radio telephone systems with prioritized and nonprioritized handoff procedures, IEEE Transactions on Vehicular Technology 35(3) (1986) 77–92.

[6] Y.-B. Lin and W. Chen, Call request buffering in a PCS network, in: IEEE Infocom (June 1994) pp. 585–592. [7] Y.-B. Lin and W. Chen, Impact of busy lines and mobility on cell blocking in a PCS network, IEEE J. Selected Areas in Communications, submitted. [8] Y.-B. Lin, S. Mohan and A. Noerpel, PCS channel assignment strategies for handoff and initial access, IEEE Personal Communications Magazine 1(3) (1994). [9] B. L. Miller, A queueing reward system with several customer classes, Management Science 16(3) (1969). [10] P. Nain and K. Ross, Optimal multiplexing of heterogeneous traffic with hard constraint, in: Proceedings of ACM Sigmetrics (1986) pp. 100–108. [11] S.-H. Oh and D.-W. Tcha, Prioritized channel assignment in a cellular radio network, IEEE Transactions on Communications 40(7) (1992) 1259–1269. [12] E. C. Posner and R. Guerin, Traffic policies in cellular radio that minimize blocking of handoff calls, ITC-11, Kyoto (1985). [13] S. Ross, Applied Probability Models with Optimization Applications (Holden-Day, 1970). [14] S. Tekinary and B. Jabbari, Handover and channel assignment in mobile cellular networks, IEEE Communications Magazine (November 1991) 42–46. [15] S. Tekinary and B. Jabbari, A measurement based prioritization scheme for handovers in cellular and microcellular networks, IEEE J. Selected Areas in Communications (October 1992). [16] J. Walrand, An Introduction to Queueing Networks (Prentice-Hall, Englewood Cliffs, NJ, 1988). [17] C. H. Yoon and K. Un, Performance of personal portable radio telephone systems with and without guard channels, IEEE J. Selected Areas in Communications 11(6) (August 1993) 911–917.

Ramachandran Ramjee received his B.Tech. in computer science and engineering from the Indian Institute of Technology, Madras in June ’92. He received his M.S. in computer science in December ’93 from University of Massachusetts, Amherst where he is currently a Doctoral student. His current research interests are signaling, mobility management and quality of service issues in wireless and high speed networks. E-mail: [email protected]

Ramesh Nagarajan received his MSEE and Ph.D. in EE from the University of Massachusetts in Amherst in 1993. He has been working in the Performance Analysis Department at Bell Labs since 1993. He is a member of IEEE and his interests are in broadband network protocols and traffic management, transport and service network design and wireless networks. E-mail: [email protected]

Don Towsley received the B.A. degree in physics and the Ph.D. degree in computer science from University of Texas in 1971 and 1975 respectively. From 1976 to 1985 he was a member of the faculty of the Department of Electrical and Computer Engineering at the University of Massachusetts, Amherst. He is currently a Professor of Computer Science at the University of Massachusetts. During 1982–1983, he was a Visiting Scientist at the IBM T. J. Watson Research Center, Yorktown Heights, NY, and during the year 1989–1990, he was a Visiting Professor at the Laboratoire MASI, Paris, France. His research interests include high speed networks and multimedia systems. He is currently on the editorial boards of Networks and Performance Evaluation and was previously on the editorial boards of IEEE Transac-

R. Ramjee et al. / On optimal call admission control in cellular networks tions on Communications, IEEE/ACM Transactions on Networking and the Journal of Discrete Event Dynamic Systems. He was a Program Cochair of the joint ACM SIGMETRICS and PERFORMANCE ’92 conference. He was elected Fellow of the IEEE for “contributions in the

41

modeling and analysis of computer networks”. He is also a member of the ACM, ORSA and IFIP Working Groups 6.3 and 7.3. E-mail: [email protected]