A Finite Buffer Queue with Priorities
A Finite Buffer Queue with Priorities Vinod Sharmaa and Jorma T. Virtamob a Deptartment of ECE, Indian Institute of Science, Bangalore-560012, India. b
Laboratory of Telecommunication Technology, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Finland. Abstract We consider a queue with finite buffer where the buffer size limits the amount of work that can be stored in the queue. The arrival process is a Poisson or a Markov modulated Poisson process. The service times (packet lengths) are iid with a general distribution. Our queue models the systems in the Internet more realistically than the usual M/GI/1/K queue which restricts the number of packets in the buffer rather than the buffer content (the number of bits). We obtain the stability, the rates of convergence to the stationary distribution and functional limit theorems for this system. In addition we also obtain algorithms to compute the stationary density of the workload process, the waiting times and the probability of packet loss. Next we study the queue with two priority classes. The higher priority traffic has preemptive-resume priority. For sharing the buffer we consider two cases. In the first case the buffer is shared by both the classes without any priority. In the second case the buffer is partitioned into two groups one reserved for each class. For this system also we obtain all the results mentioned for the single class traffic. Keywords: Finite buffer queue; MMPP arrival process; Priority classes
1
Introduction
In integrated services networks different QoS can be provided by specifying different space and time priority to the customers. For example, by providing higher time priority to a customer one can lower its time delays. But by providing higher space priority to another customer, we can lower its probability of loss even though, lower time priority means that the delays of these packets can be longer. Thus different customers can be provided differentiated services in a scalable manner. Therefore these priority schemes are being considered in the Integrated Services as well as Differentiated Services Architectures in the Internet (see Braden et al. [4], Blake et al. [3] and Ferguson and Hudson [6]). In this paper we consider a queue with a finite buffer and two priorities of traffic. In contrast to other studies, by a finite buffer of size s we mean that at most s amount of work (e.g. bits) can be stored in the queue at any time. Thus, if a packet of size x arrives but the 1
space available in the buffer is less than x then that packet will be lost. Usually, by a finite buffer one means that at most (say) K packets can be stored in the buffer. If the packet size is fixed (e.g. in ATM networks) then the two notions of buffer size are (almost) the same; otherwise they are not. Thus in the Internet our notion of buffer size is closer to reality. Since the results for our finite buffer queue even with a single class of traffic do not seem to be available, (see however Tan and Knessel [18] for the analysis of the unfinished work for a single class, Poisson arrivals and exponential service times) we will first provide the results for this case. We will compare this queue with the usual M/GI/1/K queue and show that the behaviour of the two queues can be substantially different and hence one should also study the queue with our notion of a buffer length. Interestingly, the traditional method of studying an M/GI/1/K queue at the departure epochs (see e.g., Cooper [5], Chapter 5.9) does not provide a simple way to analyze our queue. This is because the probability of admission of a new arrival depends upon the actual packet lengths of the packets stored at that moment and not just the queue length. However we show that rate conservation equations for the workload process do provide a simple feasible way for this queue. In fact, the workload process is a continuous state space, continuous time Markov process even in the case of a finite buffer queue. Our equation is a generalization of the Tak´acs integrodifferential equation. We will show that our method is effective in solving even the two priority problem. We will also use this method to obtain the results when the arrival process is generalized to a Poisson process modulated by a Markov chain (MMPP). For the two priority system, we will consider two cases. In the first case one of the classes has higher time priority (preemptive resume) but the buffer is completely shared. In the second case, the buffer is completely partitioned into s1 and s0 sizes with s = s0 + s1 although one of the classes has higher time priority. Both of these buffer sharing schemes have been considered in the literature for the usual finite buffer queues. Also, preemptive-resume time priority is a common priority scheme and is being considered in the PPP multilink fragmentation protocol for Internet by IETF. For all the systems considered (with one as well as the two class traffic) we obtain the stability, the rates of convergence and the algorithms to compute the stationary probability of loss of each class, the waiting time distributions and the busy period distribution. Now we mention the other relevant literature. The extensive study of Tan and Knessel [18] was already cited. Infinite buffer priority queues with Poisson arrivals are considered in detail in Jaiswal [7]. A single class queue with Markovian arrivals and infinite buffer is widely studied (see Neuts [12] and Ramaswami [13] for various results and references). Lin and Silvester [11] consider a two class discrete queue with iid traffic and finite buffers. They 2
obtain the probability of packet loss for both the priority classes for various buffer sharing schemes. Some other studies for discrete priority queues are Kawahara et al. [9], Takahashi and Hashida [17] and Sharma and Gangadhar [16]. Wagner [19] considers a continuous queue with Markov modulated arrivals and infinite buffers and obtains the moment generating function of the waiting times for both the priority classes. The delays for a f inite buffer queue with priority and iid and Markov modulated arrivals are considered only in Sharma and Gangadhar [16]. However the methods used in this paper are necessarily very different. The paper is organized as follows. In Section 2 we study the system with a single class of traffic. In Section 3 we will study the two class problem where the two classes have separate buffers to store their packets. The system with a single buffer shared by both the classes is studied in Section 4. In each section we will provide specific algorithms for the stationary waiting times of admitted packets and the probability of packet loss of each class. Some computational results will also be presented and all the results will be generalized to the MMPP arrival traffic.
2
Single class traffic
In Section 2.1 we consider a finite queue with a Poisson arrival process. In Section 2.2 we will generalize our results to the MMPP case.
2.1
Poisson arrivals
We consider the M/GI/1/s-queue with a finite buffer of size s. Let λ be the Poisson arrival rate and the packet sizes be iid with a generic packet length X. The server serves at a unit rate. Also let the total workload (the unfinished work) at time t be Vt . The buffer space is assumed to be allocated dynamically, i.e. when a packet of size X arrives, the packet is admitted into the buffer whenever the total free space in the buffer (= s − Vt ) is greater than or equal to X. In this case, the finite buffer size does not set a constraint on the number of packets but on the total workload (e.g. bits) in the buffer. Since this queueing system has not been studied much before, first we study its tability and the rates of convergence to the stationary distribution in Section 2.1.1. In Section 2.1.2 we will provide the algorithms for the distributions of the workload process, the waiting times and the probability of packet loss. In Section 2.1.3 we will study the busy period.
3
2.1.1
Stability
The process {Vt } is Markov with state space [0, s]. It is also regenerative where we can (and will) take the regeneration epochs as the arrival epochs that find the system empty. Let t = 0 be the arrival epoch of the first customer arriving to the system which finds the system empty and let τ be the number of the next arrival finding the system empty. Since Vt ≤ s at all times, the probability that an arrival will find the system empty is lower bounded by ∆
p1 = P [interarrival time > s] = e−sλ > 0. Thus P [τ > n] ≤ (1 − p1 )n and hence τ has all finite moments and also finite exponential moments in a neighbourhood around zero. If {an } are interarrival times and τˆ is a regeneration length for {Vt } then P τˆ is distributed as τk=1 ak and hence τˆ also has all finite moments and finite exponential moments in a neighbourhood of zero. Furthermore, τˆ has a spread-out distribution. Thus by regenerative process theory (see Asmussen [1], Chapter V) {Vt } has a unique stationary distribution π and the process converges to it in total variation starting from any initial conditions. Also, E[ˆ τ α ] < ∞ for any α ≥ 1 implies (see Kalashnikov [8]) that supA |P [Vt ∈ A] − π(A)| < ct1−α for some constant c. Similarly, from finite exponential moments, we obtain exponential rates of convergence. Furthermore Vt ≤ s and E[ˆ τ α ] < ∞ also imply hR α i τˆ that E < ∞ for any α ≥ 0. Now using Sharma [15], Asmussen [1], we obtain 0 Vt dt functional CLT and LIL theorems for {Vt }. Next we consider other processes of interest. Let Wn be the waiting time of the nth customer. The arrival epochs finding the queue empty are regeneration epochs for {Wn } also with the regeneration length τ . Since P [τ = 1] ≥ p1 , the distribution of τ is aperiodic. Thus, again from regenerative theory, {Wn } has a unique stationary distribution (say π) and sup |P [Wn ∈ A] − π(A)| < C1 n1−α A
for some constant C1 < ∞ and any α > 0. The functional limit theorems hold in the same way. Now consider the queue length process {Lt } where Lt is the number of customers in the system at time t. This process is not necessarily a finite state process. Again τˆ is the regeneration length for {Lt } and hence uniqueness and existence of stationary distributions and rates of convergence to π hold. To show the finiteness of moments of Lt , observe that during a regeneration cycle Lt is upper bounded by the number of customers served in that cycle and hence we have Eπ [Lαt ] < ∞ for all α. 4
In the next subsection we will consider the arrival process to be MMPP. Then we will provide the necessary changes in the arguments to extend these results to the MMPP case. In Section 4 we will be considering the queue with two classes of traffic. Then we will be considering the processes {Vt (i)}, {Wn (i)}, {Lt (i)} for class i, i = 0, 1. Again τ will denote the regeneration length for {(Wn (1), Wn (0))} (any arrival epoch finding the system empty will be a regeneration epoch if we probabilistically assign the class which we always can for our model). These regeneration epochs will remain the same if we consider a single class system with arrival rates λ1 + λ0 and the service distribution as the mixture of the service distributions of the two classes (with probabilities λ1 /(λ1 + λ0 ) and λ0 /(λ1 + λ0 )). Thus, from the single class result we also obtain E[τ α ] < ∞ and E[ˆ τ α ] < ∞ for the two class problem and all the above results carry over directly. The system in Section 3 requires a slight modification to the above arguments. We will mention them there.
2.1.2
Workload distribution
Our task in this subsection is to determine the stationary distribution F of the workload (unfinished work) V in the buffer, the packet loss probability p, the traffic loss probability p(t) (the fraction of number of bits lost) and the waiting time distribution. Due to the PASTA property, p = P{X + V > s}. Note that p and p(t) are two different quantities, because the loss process depends on the packet size. First we establish the existence of density f (x) of F (x) for x > 0. Let A be a Borel subset of IR such that 0 6∈ A and Lebesgue measure of A is 0. From the dynamics of the queue, R R for any initial distribution, 0∞ 1{Vt ∈A} dt = 0 a.s. Therefore, π(A) = Eπ [ 0τ 1{Vt ∈A} dt]/E[τ ] = R
R
Eπ [ 0∞ 1{Vt ∈A} 1{τ >t} dt]/E[τ ] ≤ Eπ [ 0∞ 1{Vt ∈A} dt]/E[τ ] = 0. Thus F is absolutely continuous with respect to Lebesgue measure on {x : x > 0}.
We use a Tak´acs type approach to compute F . Let B(x) and b(x) be the cdf and the pdf of the packet size X. Consider the balance of probability flows across a control surface at the workload level x under stationarity. The downward flow, due to the emptying of the buffer, is equal to the probability density at x, f (x) = F 0 (x). The upward flow is due to arrivals. An arrival causes the system state to cross the control level at x provided that the buffer content before the arrival, denoted by y, is less than or equal to x and that the size of the arriving packet is greater than x − y (in order to cross the control level) but less than
5
s − y (in order for the packet to be admitted into the buffer). Thus we obtain
Z
f (x) = λ P0 Q(x, 0) +
0
x
Q(x, y)f (y)dy ,
(1)
where Q(x, y) = B(s − y) − B(x − y) and P0 is the (so far unknown) probability of an empty queue. In this form, written in terms of the pdf function f (x), the Tak´acs equation is an integral equation, a Volterra equation of the second kind (in contrast to an integro-differential equation; see Kleinrock [10], Chapter 5.12, pp. 226-230). We note from the equation that the value of P0 is just a scaling factor, i.e. the solution to this equation is proportional to P0 . Then, it is sufficient that we solve the equation for the case P0 = 1. From now on we let φ(x) denote the solution corresponding to this case; the solution for any value P0 is then f (x) = P0 φ(x). Let us further denote Z
I=
s
0
φ(x)dx.
In order to get a distribution with unit probability we must have P0 + P0 I = 1
(2)
from which
1 . 1+I With P0 known, the distribution of the unfinished work is completely determined. It consists of an atom P0 at zero and of a continuous distribution with the pdf P0 φ(x) in the interval (0, s). P0 =
Let us take the mean packet size m = E[X] as the unit of work and denote by ρ = λm the (offered) load of the system. The admitted load ρ0 is equal to 1 − P0 . Thus the fraction of traffic lost is p(t) = (ρ − ρ0 )/ρ = 1 −
I/ρ . 1+I
(3)
The packet loss probability is
p = P0 1 − B(s) +
Z 0
s
(1 − B(s − y))φ(y)dy .
(4)
The Volterra equation (1), with P0 = 1, can be solved in a straightforward way starting from the value at x = 0 and proceeding to greater values. First, note that f (0) is directly determined by the equation, f (0) = λ Q(0, 0) = λ B(s). 6
Second, for any x the value of f (x) is determined by the values of f (y) for y ≤ x. In particular, consider a numerical scheme, where we wish to determine the values of f (x) at the grid points xi = i∆ for i = 0, . . . , n with s = n∆. Consider now Eq. (1) at xi . Whatever numerical scheme is used for the evaluation for the integral, the value of the integral is a linear combination of the values of f (xj ) for j = 0, . . . , i. Thus, (1) is a linear equation for f (xi ) which can be solved in terms of the previous values of f (xj ), j = 0, . . . , i − 1. So starting from i = 0 we can solve the values consecutively for greater values of i up to n. We have solved the Volterra equation as described above, using Simpson’s rule for the integration (with a slight modification for an odd number of intervals), and calculated the traffic and packet loss probabilities p(t) and p from equations (3) and (4) for an M/M/1/s system with the buffer size s = 3 (in units of mean packet size). The loss probabilities are plotted as a function of the offered load ρ in Fig. 1. For comparison we have also plotted the curve for an ordinary M/M/1/K system with K = 3 buffer places (including the buffer space of the server). In this case the packet and traffic loss probabilities are equal and are given by ρK /(1 + ρ + · · ·+ ρK ). Of course as s → ∞ the stationary distribution of each queue tends to that of M/N/1 queue. This assertion holds for the general service time distribution also. 0.6
loss probability
0.5 traffic loss 0.4
loss with 3 buffer places
0.3 0.2
packet loss 0.1 0 0.25
0.5
0.75
1
1.25
1.5
1.75
2
offered load
Figure 1: Packet and traffic losses for a finite queue with exponential packet size distribution. Observe the rather high level of p(t) for low loads. When the load approaches zero, each arriving packet finds the system empty. Then the probability that a packet is not admitted is P [X > s] = e−s . The average size of a lost packet E[X|X > s] is (s + 1) resulting in the traffic loss probability p(t) ≈ E[X|X > s]P [X > s] = (s + 1)e−s . For s = 3 the value is 7
numerically around 4 · 0.05 = 0.2 as shown in the figure. The finite loss probabilities at low loads in the above example (as well as in some later examples) are due to the assumed exponential packet size distribution. It would be more realistic to assume the packet size to be bounded. This poses no problem for the numerical solution, and we demonstrate this by another example shown in Fig. 2. Here the packet size distribution is Exp(1) truncated at 1, with mean 0.418. The buffer size is 1.254, i.e. three times the mean. In this case the blocking probabilities tend to zero at low loads, since the buffer can always hold at least one packet. 0.6 loss with 3 buffer places
loss probability
0.5 0.4 0.3 traffic loss 0.2
packet loss 0.1 0 0.25
0.5
0.75
1
1.25
1.5
1.75
2
offered load
Figure 2: Packet and traffic losses for a finite queue with truncated exponential packet size distribution. By PASTA, under stationarity the workload seen by an arriving customer also has the cdf F and the pdf f . Then the density of the stationary waiting time of an admitted packet is (for x > 0) P [V ∈ (x, x + dx)|a packet is admitted] =
P [a packet is admitted|Vt ∈ (x, x + dx)]f (x)dx P [a packet is admitted]
= f (x)B(s − x)dx/(1 − p). Also the probability that an admitted packet does not wait is P0 B(s)/(1 − p). Similarly, the sojourn time of an admitted packet has density Z 0
x
f (y)b(x − y)dy/(1 − p) + P0 b(x)/(1 − p).
8
2.1.3
Busy periods
We will need the distribution of the busy periods of the queue in later sections for calculation of the waiting times of the lower priority packets. Therefore, we compute these quantities now. We denote by Tb (x) the busy period if the period starts (say at time t = 0) with initial workload of x. We are not assuming that there is an arrival at t = 0 and hence Tb (0) = 0. Let S(x) be the work that enters the queue during time [0, x]. We will calculate the distribution of S(x) later on. Then, we can solve the following equation satisfied by E[Tb (x)] Z
E[Tb (x)] = x +
s 0
E[Tb (y)]dP [S(x) = y].
(5)
To solve this equation numerically we discretize x to x0 = 0 < x1 < · · · < xn = s. Then for E[Tb (xi )], i = 0, . . . , n, we get a system of (n + 1) linear equations in (n + 1) variables. Observing that E[Tb (0)] = 0, we can then solve the other n unknowns from the resulting n equations. Similarly for E[Tb (x)2 ] we get the integral equation E[Tb (x)2 ] =
Z
x
0
E[(x + Tb (y))2]dP [S(x) = y]
= x2 + 2x
Z
0
s
Z
E[Tb (y)]dP [S(x) = y] +
s
0
E[Tb (y)2]dP [S(x) = y].
(6)
Now from the distribution of S(x) and the solution of (5), we can solve for (6). Higher moments of Tb (x) can be obtained in the same way. Furthermore, the density fx of Tb (x) satisfies the equation
Z
fx (y) =
s
0
fz (y − x)dP [S(x) = z]
where y ≥ x. This equation can be solved by discretizing the variables x, y and z. If T0 denotes a busy period started by an arriving packet, its moments (and distribution) can be obtained from that of Tb (x) using the relation E[T0α ]
Z
=
0
s
E[Tb (x)α ]b(x)dx.
Now we obtain the distribution of S(x) using the rate equations for transient probabilities. We have, for h > 0, P (Vt+h ≤ y) = P (Vt ≤ y + h)(1 − hλ) + hλ
Z 0
y
ft (z)[B(y − z) + (1 − B(s − z))]dz + o(h)
where ft is the pdf of Vt given V0 = x. Subtract P (Vt ≤ y), divide by h and let h go to zero. Using P (Vt = 0) = 0 for 0 ≤ t < x we get by integrating over time from 0 to T ≤ x Z 0
y
(fT (z) − f0 (z))dz =
Z 0
T
(ft (y)dt − ft (0))dt − λ 9
Z 0
T
Z 0
y
ft (z)Q(y, z)dzdt.
By discretizing for y, z and t we can solve this equation and then fx provides the density of S(x). Observe that there is one equation for each pair of discretized values of y and T . Thus there are as many equations as there are unknowns variables ft (y) at the discretized points. Using similar rate equations for transient probabilities, we can also compute the distribution of Tb (x).
2.2
MMPP arrivals
Consider the case where the input process is MMPP with an n-state modulating process {Yt }. Let {Yt } have the transition rate matrix with elements qi,j . Let λi denote the Poisson arrival rate and Bi (t) the distribution of the arriving packet lengths when {Yt } is in state i. Now we obtain the existence, uniqueness, rates of convergence etc. results for this system. Define λ = max λi . Then, whenever an arrival comes with an interarrival time greater than s (this happens with probability lower bounded by e−λs > 0), the system is empty. Thus, if τ denotes the number of the first arrival (if the system starts at t = 0 with an arrival to an empty system) to the empty system, we obtain, as in the last subsection, E[τ α ] < ∞ for all α ≥ 0. However, these epochs are not the regeneration epochs for this system. Now consider a state i0 for {Yt } which is assumed to be reachable from any other state. We will consider an arrival epoch to the empty system when Yt = i0 as a regeneration epoch. Then, because of finite state space there is a 0 < ∆ < s and a δ > 0 such that P [Y∆ = i0 |Y0 = i] > δ > 0 for all i. Hence, if an arrival comes to an empty system then a time s after it, with probability lower bounded by δe−(qi0 ,i0 +λ)s the system is empty and Yt is in state i0 . Therefore, with probability lower bounded by δe−(qi0 ,i0 +λ)s e−qi0 ,i0 δ1 (1 − e−λi0 δ1 ) within time s + δ1 , an arrival comes to the empty system and {Yt } is in state i0 . Therefore denoting by τ˜ the regeneration length (in terms of number of arrivals) and τˆ the regeneration length of the process {(Vt , Yt )}, we obtain E[˜ τ α ] < ∞ and E[ˆ τ α ] < ∞ for any α > 0. Finiteness of exponential moments in a neighbourhood of 0 can also be shown. Thus we obtain uniqueness, existence of the stationary distribution of {(Vt , Yt )} as well as rates of convergence and function limit theorems. Similarly we can obtain the results for the queue length and the waiting time process. Next we obtain algorithms to compute the stationary density of (Vt , Yt ), the waiting time distributions and the probability of packet loss.
10
Define Fi (x) = P [V ≤ x, Yt = i] and let fi (x) be the density of Fi (x) for x > 0. In the same way as before we can write the balance equation for the stationary system. Consider the set of states {V ≤ x, Yt = i} and write the balance of probability flows across a control surface enclosing this set of states. This gives Z Fi0 (x) +
X j
qj,iFj (x) = λi
x
0
Qi (x, y)dFi(y).
where Qi (x, y) = Bi (s − y) − Bi (x − y). In terms of the pdf’s, extracting the atoms at 0, this becomes
Z
fi (x) = λi Pi Qi (x, 0) +
x
0
Qi (x, y)fi(y)dy −
X j
qj,i Pj +
Z 0
x
fj (y)dy ,
(7)
where Pi = Fi (0). Now Eq. (7) can be solved in the same way as before, proceeding from 0 to greater values of x: fi (x) depends on the values of fj (y), j = 1, . . . , n for y ≤ x only. The essential difference is that now we have several unknowns, Pi , i = 1, . . . , n. Suitable boundary conditions have to be formulated in order to solve these values. Since the buffer content is always ≤ s we have Fi (s) = πi , where the πi are the steady state probabilities of Yt . Denote Z s Z s I = ( f1 (x)dx, . . . , fn (x)dx). 0
0
Now we can solve Eq. (7) for the specific initial condition Pi = 1 and Pj = 0 for j 6= i. Let Ii be the value of the vector I resulting from these initial conditions. Then the Pi can be solved from the equation P+
X i
Pi Ii = π,
where P = (P1 , . . . , Pn ). This generalizes the normalization condition (2) for the Poisson input case. Next we consider the stationary probability of packet loss and the waiting times of the admitted packets. For this we need to compute the stationary probability of {Vt } seen by an arriving packet. This is the Palm probability corresponding to the time stationary probability of {(Vt , Yt )} computed above and hence can be computed via the general formulae (see e.g. Baccelli and Bremaud [2]). However, for the MMPP case we can actually have a simple direct formula which we now provide. Let N(a, b) be the number of arrivals under stationarity to the system during the time interval (a, b). Then lim P [Vt ≤ x, Yt = i|N(t, t + h) = 1] h↓0
11
P [N(t, t + h) = 1|Vt ≤ x, Yt = i]P [Vt ≤ x, Yt = i] h↓0 P [N(t, t + h) = 1] λi Fi (x) . = P j πj λj
= lim
Thus the probability of packet loss p is p=
X
P
i
λi [Pi (1 − Bi (s)) + j πj λj
Z
s
0
fi (x)(1 − Bi (s − x))dx]
and the probability that an admitted packet has to wait time x > 0 (similarly one can write for x = 0) has the density X λi fi (x) Bi (s − x) . P j πj λj (1 − p) i Of course now the sojourn time distribution can also be computed. We illustrate the computation of the probability of packet loss, the fraction of traffic lost P (= 1 − (1 − Pi )/ρ) and the pdf of the stationary waiting times via the above equations by an example. Consider a queue with {Yt } having two states with transition rates q1,2 = 0.1, q2,1 = 0.3, λ1 = 0.5, λ2 = 2.5 and service times exponentially distributed with mean 1. The probability of packet loss is shown in Fig. 3 and the waiting time density in Fig. 4 (it has an additional atom at 0 of mass 0.789). 0.6
loss probability
0.5 traffic loss 0.4 0.3 packet loss 0.2 0.1 0 0.25
0.5
0.75
1
1.25
1.5
1.75
2
offered load
Figure 3: Probability of packet and traffic loss in MMPP queue. To compute the waiting time distribution of the lower priority packets in Section 3.2, we will need the distribution of Tb (i, x), the first time the queue becomes empty if started at t = 0 in state (i, x). The moments and distribution of Tb (i, x) can be computed as for the Poisson case with slight changes. We show it by calculating the joint distribution of Tb (i, x) 12
0.25
pdf
0.2 0.15 0.1 0.05 0 0.5
1
1.5
2
2.5
3
waiting time Figure 4: Waiting time density for MMPP queue. and YTb (i,x) which is obtained by solving the following simultaneous equations XZ
P [Tb (i, x) = y, YTb(i,x) = j] =
i1
P [Vx ∈ (x1 , x1 + dx), Yx = i1 |V0 = x, Y0 = i]
·P [Tb(i1 , x1 ) = y − x, Yy−x = j]. By discretizing the variables x, y we can solve these equations if we also know P [Vx ∈ (x1 , x1 + dx), Yx = i1 , |V0 = x, Y0 = i]. One way to obtain this transition matrix is by solving the rate equations corresponding to the transient probabilities. To obtain these, observe that for h > 0 small, P [Vt+h ≤ y, Yt+h = j] = (1 − hλ Z j − hqj,j )P [Vt ≤ y + h, Yt = j] y
+hλj +
X
k6=j
0
P [Vt = z, Yt = j](Bj (y − z) + (1 − Bj (s − z))dz
hqk,j P [Vt ≤ y + h, Yt = k] + o(h).
Now subtract P (Vt ≤ y, Yt = j) from both sides, divide by h and let h tend to 0. Since the state Vt = 0 will not be visited till time Tb (i, x), we obtain, for T ≤ x Z
y 0
(fT,j (z) − f0,j (z))dz =
Z 0
+
T
ft,j (y)dt − λj
X k
Z
qk,j
0
T
Z 0
y
Z 0
T
Z 0
y
ft,j (y 0 )Qj (y, y 0)dy 0
ft,k (y 0)dy 0dt.
This system of equations can be solved as in the last section except that we have to solve them simultaneously.
13
3
Two priority classes with completely partitioned buffer
3.1
Poisson arrivals
In this section we consider a system with two classes. Class 1 traffic has preemptive-resume priority over class 0 traffic, i.e. whenever there is no class 1 traffic in the system the remaining service of class 0 packets resumes. There is a separate buffer for each class. The arrival process of each class is Poisson and the service times are iid with general distributions. For class i the arrival rate is λi , a generic packet length is X(i) with cdf Bi and pdf bi and its buffer can store a total work of si at a time. Let Vt (i) be the buffer content at time t in the buffer of class i. The existence and uniqueness of stationary distributions of (Vt (1), Vt (0)), the waiting times and the queue length process follow in the same way as in Section 2. Similarly, the rates of convergence hold. Indeed observe that we can form this system by considering a single Poisson stream with arrival rate λ1 + λ0 . On arrival a packet is made of class 1 with probability λ1 /(λ1 + λ0 ); otherwise of class 0. Then any arrival epoch finding the system empty will be a regeneration epoch. Since this can happen at any arrival time with probability lower bounded by e−(s1 +s0 )(λ1 +λ0 ) we can now repeat the arguments for the single class case. Now we will provide algorithms to compute the density of the stationary distribution of (Vt (1), Vt (0)) via the rate conservation equations. Then we will compute the stationary probability of packet loss, and the waiting time distributions of each class. We will obtain the rate conservation equations in several steps. Also, we will denote the density of P [0 < V (1) ≤ x, V (0) = 0] by g(x), the density of P [V (1) = 0, 0 < V (0) ≤ y] by h(y), the density of P [0 < V (1) ≤ x, 0 < V (0) ≤ y] by f (x, y) and the stationary probability that the system is empty by P0 . The existence of the respective densities can be shown by an argument similar to the one used in subsection 2.1.2 for the one-dimensional single class case. The rate equation at (V (1) = 0, V (0) = 0) is g(0) + h(0) = P0 (λ1 B(s1 ) + λ0 B0 (s0 )).
(8)
Similarly the rate equation for the set (0 < V (1) = x, V (0) = 0) is Z
g(x) + P0 λ1 B1 (x) = g(0) + λ1
0
x
0
0
0
g(x )Q1 (x, x )dx + λ0 B0 (s0 ) 14
Z 0
x
g(x0 )dx0 ,
(9)
where Qi (x, x0 ) = Bi (si − x0 ) − Bi (x − x0 ), i = 0, 1. (This equation is illustrated in Fig. 5 where each arrow corresponds to one term in the equation). By discretization we can solve for g(xi ), xi > 0 in terms of g(0). Finally, we can obtain g(0) by observing that g(s1 ) = 0. Next consider the rate equation for the set (V (1) = 0, 0 < V (0) ≤ y) Z
h(y) + P0 λ0 B0 (y) +
0
y
f (0, y 0)dy 0 Z
= h(0) + λ1 B1 (s1 )
y
0
Z
h(y 0)dy 0 + λ0
y
h(y 0 )Q0 (y, y 0)dy 0.
0
(10)
This equation can again be solved for h(yi ) by discretization for successively larger values yi if we also know h(0) and f (0, y 0), 0 < y 0 < y. The initial value h(0) can be obtained from (8) by plugging in g(0) obtained above (it is still in terms of the unknown P0 to be found later on). To get f (0, y 0) we need the rate equations for the strip (0 < V (1) ≤ x, V (0) ∈ (y, y + dy)), which are Z
f (x, y) + h(y)λ1 B1 (x) + λ0 b0 (y) Z
= f (0, y) + λ1
x
0
x
0
0
Z
0
g(x )dx + λ0
x
0
0
dx
f (x0 , y)Q1(x, x0 )dx0 + λ0 B0 (s0 − y)
Z
0
Z
x 0
y
f (x0 , y 0)b0 (y − y 0 )dy 0
f (x0 , y)dx0.
(11)
We illustrate this equation in Fig. 5. Finally we obtain g(xi), h(yi ), f (xi , yi) in terms of P0 which is then obtained from the normalization equation Z
P0 +
0
s1
Z
g(x)dx +
s0
0
Z
h(y)dy +
0
s1
Z 0
s0
f (x, y)dxdy = 1.
These equations can be solved by discretizing x and y variables by keeping y fixed for various values of x if we know h(y) and f (0, y) and f (x0 , y 0), y 0 < y. We will obtain f (0, y) by fixing f (s1 , y) = 0. Then f (0, y) can be used to obtain h(y 00 ) for y 00 > y from Eq. (10) which again will be used in (11) to obtain f (x, y 00 ) and so on. The initial conditions of f (x0 , y1) will be obtained from (11) by observing that the integral containing f (x0 , y 0) can then be made zero. The above discretization can be used to approximate f (x, y). The finer the discretization is, the better is the approximation. We use this approximation to obtain other quantities of interest. Consider the class 1 traffic. This traffic is not affected by the class 0 traffic. Therefore, the probability of packet loss p(1) of class 1, and the waiting time distribution are obtained as for the single class traffic in the last section. Now consider the class 0 traffic. By PASTA, on arrival a class 0 packet sees the workload of y in its queue with the probability h(y) +
R s1 0
∆
f (x, y)dx = f0 (y). Then the probability p0 that a class 0 packet is lost is (1 − B0 (s0 ))(P0 +
Z 0
s1
Z
g(x)dx) + 15
0
s0
f0 (y)(1 − B0 (s0 − y))dy.
y
y
s0
s0
s1
x
s1
x
Figure 5: Illustration of the probability rate flows in Eq. (9) (on the left) and Eq. (11) (on the right). Also, the rate of the work entering the system is ρ0 = 1 − P0 . The fraction of class 0 traffic lost is
Z
(P0 +
0
s1
Z
g(x)dx)
∞
s0
Z
zb0 (z)dz +
0
s0
Z
(h(y) + f (x, y))
∞
s0 −y
zb0 (z)dz /E[X(0)].
Similarly one can write the fraction of class 1 traffic lost. We illustrate the computation of the above equations by an example. We take λ0 = 1.5λ1 , the mean service rate 1, and all the distributions are exponential. The two buffer sizes are s0 = 5 and s1 = 3. Fig. 6 provides the probabilities of the packet loss and the traffic loss as a function of λ1 . Fig. 7 gives the plot of f (x, y) and Fig. 8 of g(x) and h(y) when λ1 = 0.2. 0.5
loss probability
0.4 0.3
class 0, traffic loss
class 1, traffic loss class 0, packet loss
0.2 0.1
class 1, packet loss
0 0.2
0.4
0.6
0.8
1
offered load of class 1 traffic
Figure 6: Packet loss probability for the two classes. Next we compute the waiting time distribution of a class 0 packet (till the time its service first starts) that is admitted. By PASTA, an arriving class 0 packet sees the workload 16
5 4 3 2 1 0.05 0.04 0.03 0.02 0.01 0 0 1 2 3
Figure 7: Waiting time density for x > 0, y > 0. (Vt (1) = x, Vt (0) = y) with stationary density f (x, y) (or g(x) or h(y) as the case may be). Thus the stationary density seen by an admitted class 0 packet is f (x, y)B0(s −y)/(1 −p(0)). Now we compute the waiting time of an admitted class 0 packet (denote its arrival time as time 0) conditioned on it seeing the work (x1 , y1 ) on its entrance. The first time the class 1 queue will be empty will be denoted by τ (x1 ) which is independent of y1 and its distribution and moments can be computed as for a single class queue in Section 2. Let the first arrival of class 1 after time τ (x1 ) that is admitted occur at time τ (x1 ) + I1 . I1 is exponentially distributed with parameter λ1 B1 (s1 ). During time [τ (x1 ), τ (x1 ) + I1 ) class 0 packets can be served. If I1 > y1 then the service of class 0 packet under consideration will start during this time and its waiting time will be τ (x1 ) + y1 . Otherwise this packet will have to wait for the next time when class 1 queue is empty. Let that time be τ (x1 ) + I1 + τ1 . Let {τn } be iid with the distribution of τ1 (which is known from Section 2) and let {In } be iid with the distribution of I1 . Let N = inf{n : I1 + I2 + . . . + In > y1 }. Then N − 1 is Poisson distributed with parameter λ1 B1 (s1 )y1 . The waiting time distribution of the packet under consideration is τ (x1 ) +
N −1 X i=1
τi + y1 ,
where τ (x1 ) is independent of {τn }. Thus, we can calculate the distribution (and moments) of the waiting times of the packet. 17
0.2
0.15
0.1
h(y) g(x)
0.05
0 1
2
3
4
5
x,y
Figure 8: Waiting time density for x = 0, or y = 0.
3.2
MMPP arrivals
Consider a finite state Markov chain {Yt } with transition rates qi,j which modulates the arrival process in the following way. If during time [t, t + h], {Yt } is in state i then during this interval class j, j = 0, 1 arrivals occur according to a Poisson process with rate λi,j and the service time distribution of these arrivals have cdf Bi,j and pdf bi,j . Now the process {(Vt (1), Vt (0), Yt )} is Markov. Also, the existence and uniqueness of stationary distributions, rates of convergence and functional limit theorems hold as for the single class case using the additional arguments of Section 3.1. Next we provide the algorithms for computing the stationary density of {(Vt (1), Vt (0), Yt)}, the waiting distributions and the probability of packet loss for each class. First we provide the rate conservation equations for {(Vt (1), Vt (0), Yt )}. We will denote the stationary density at (x, y, i), x > 0, y > 0 by fi (x, y), for x = 0, y > 0, i by hi (y) and for x > 0, y = 0, i by gi (x). The rate equation at state (0, 0, i) is gi(0) + hi (0) +
X j
qj,iPj = Pi
X
λi,k Bi,k (sk ),
k
and on the axis 0 < V (1) ≤ x, V (0) = 0, Y = i is gi(x) + Pi λ1 Bi,1 (x) + Z
= gi (0) + λi,1
0
x
0
X j
Z
qj,i
x
0
gj (x0 )dx0
0
0
gi (x )Qi,1 (x, x )dx + λi,0 Bi,0 (s0 )
Z 0
x
gi (x0 )dx0 ,
where Qi,j (x, x0 ) = Bi,j (sj − x0 ) − Bi,j (x − x0 ). Similarly, on the axis V (1) = 0, 0 < V (0) ≤ 18
y, Y = i we get Z
hi (y) + Pi λi,0 Bi,0 (y) +
y 0
fi (0, y 0)dy 0 + Z
= hi (0) + λi,1 Bi,1 (s1 )
0
y
X j
0
Z
qj,i
y
0
0
hj (y 0)dy 0
Z
hi (y )dy + λi,0
0
y
hi (y 0)Qi,0 (y, y 0)dy 0
and for 0 < V (1) ≤ x, V (0) ∈ (y, y + dy), y > 0 Z
fi (x, y) + hi (y)λi,1Bi,1 (x) + λi,0 bi,0 (y) Z
+λi,0
x
0
dx0
Z
y 0
Z
= fi (0, y) + λi,1
0
0 0
x
gi (x0 )dx0
fi (x0 , y 0 )bi,0 (y − y )dy 0 + x
X j
qj,i fj (x, y)
fi (x0 , y)Qi,1(x, x0 )dx0 + λi,0 Bi,0 (s0 − y)
Z 0
x
fi (x0 , y)dx0.
These equations are solved numerically using previous methods. To obtain the probability of packet loss and the waiting time distributions, we need the stationary probability of {Vt (1) ∈ (x, x + dx), Vt (0) ∈ (y, y + dy) and Yt = i} at an arrival P instant of class j. For x > 0, y > 0 this probability is λi,j fi (x, y)/( k λk,j πk ). Similarly one can obtain the density for x = 0, y > 0; x > 0, y = 0 and the atom at x = y = 0. Therefore, the probability of packet loss of class 1 is Z s1 Z s0 X 1 p(1) = P λi,1 gi (x) + fi (x, y)dy (1 − Bi,1 (s1 − x))dx 0 0 k λk,1 πk i Z
+
s0
0
hi (y)dy + Pi (1 − Bi,1 (s1 )) .
Similarly, one can calculate the probability of packet loss for class 0. The probability that an admitted packet of class 1 experiences a delay of x > 0 (similarly one can write for x = 0) is "
#
Z s0 X 1 1 . λi,1 fi (x, y)dy + gi (x) Bi,1 (s1 − x) P 1 − p(1) 0 k λk,1 πk i
Next we compute the waiting time distribution of an admitted packet of class 0. For this, we need the stationary system state probability as seen by an admitted class 0 packet. This can be calculated as above and the probability that it sees V (1) = x, V (0) = y, Y = i is given by (say for x > 0, y > 0) P
1 Bi,0(s0 − y) . λi,0 fi (x, y) 1 − p(0) k λk,0 πk
Let the state an admitted (say at time t = 0) class 0 packet sees be (x1 , y1, i1 ). Then, the joint distribution of the first time the class 1 packets are not in the system (denoted by 19
τ (x1 , i1 )) and the state i2 of the chain {Yt } at time τ (x1 , i1 ) can be computed as for the single class result provided in Section 2.2. Let τ (x1 , i1 ) + I1 be the time the first packet of class 1 is admitted to the queue after time τ (x1 , i1 ). Given the distribution of i2 , the distribution of I1 can be computed and also the distribution of the system at time τ (x1 , i1 ) + I1 (these can be computed by writing the rate conservation equations for transient probabilities just as at the end of Section 2.2). Now the distribution of the waiting time of the admitted class 0 packet can be computed as for the Poisson case although the actual computations are much more involved.
4 4.1
Queue with completely shared buffer Poisson arrivals
We consider the system of the last section except that now there is only one buffer of length s. The rest of the notation and assumptions are as in the last section. There is no priority for either class for storing in the buffer, i.e. if a packet of either class comes and there is space in the buffer to store the packet, the packet will be admitted. The existence and uniqueness of stationary distributions for this system were already obtained in Section 2.1. There we also obtained the rates of convergence and the functional limit theorems. Next we obtain the stationary probability density of the buffer content (Vt (1), Vt (0)), the probability of loss and the distribution of waiting times of the two classes of packets. We will follow the approach of the last section. First we will obtain the integral equations of f (x, y), g(x) and h(y) via the rate conservation equations. We will solve these equations iteratively by discretization. Then we will calculate the stationary probability of loss and waiting times. This system is more complicated than that of the last section. Now we write the rate conservation equations to compute the stationary densities f (x, y), g(x) and h(x) of (V (1), V (0)). The overall procedure and notations are as in the last section. The rate equation at (0, 0) is g(0) + h(0) = P0 [λ1 B1 (s) + λ0 B0 (s)].
(12)
The rate equation for the set {0 < V (1) ≤ x, V (0) = 0} is Z
g(0) + λ1
0
x
0
0
g(x )Q1 (x, x ) + λ0
Z 0
x
g(x0 )B0 (s − x0 )dx0 = g(x) + P0 λ1 B1 (x), 20
(13)
where Qi (x, x0 ) = Bi (s − x0 ) − Bi (x − x0 ), and for the set {V (1) = 0, 0 < V (0) ≤ y} Z
h(0) + λ1
0
y
0
0
0
h(y )B1 (s − y )dy + λ0 Z
= h(y) + P0 λ0 B0 (y) +
0
y
Z
y
0
h(y 0 )Q0 (y, y 0)dy 0
f (0, y 0)dy.
(14)
Finally, the rate equation for the strip {0 < V (1) ≤ x, V (0) ∈ (y, y + dy)} is Z
f (x, y) + h(y)λ1B1 (x) + λ0 b0 (y) Z
= f (0, y) + λ1
x
0
0
x
g(x0 )dx0 + λ0
Z
f (x0 , y)Q1(x, x0 − y)dx0 + λ0
x
Z0
dx0
x
Z
y 0
dy 0 f (x0 , y 0)b0 (y − y 0)
B0 (s − y − x0 )f (x0 , y)dx0.
0
(15)
We solve Eqs. (12)-(15) via discretization of (x, y) variables to obtain the values for g(x), h(y) and f (x, y) in exactly the same way as in the last section. We compute the probability of packet loss of each class based on the stationary probabilities computed above. By PASTA, on arrival a class i packet sees (Vt (1) = x, Vt (0) = y) work with probability density f (x, y) or g(x) or h(y) as the case may be. Thus, the probability that a class i packet is lost is Z
p(i) =
s
0
+
Z
Z
0 s
0
s
f (x, y)(1 − Bi (s − x − y))dxdy +
Z 0
s
g(x)(1 − Bi (s − x))dx
h(y)(1 − Bi (s − y))dy + P0 (1 − Bi (s)).
Also, the amount of work entering the system in unit time is given by ρ0 = 1 − P0 . By PASTA, the probability density that an admitted class 1 packet will be delayed by time x > 0 is Z 0
s−x
f (x, y)B1(s − x − y)dy + g(x)B1 (s − x) /(1 − p(1)).
Next we calculate the waiting time distribution of the class 0 packets. This is substantially more difficult to compute. Using PASTA, the stationary probability density of (Vt (1), Vt(0)) at (x, y), x > 0, y > 0 as seen by an admitted (say at time 0) class 0 packet is f (x, y)B0(s − x − y)/(1 − p(0)). Thus the probability density at (x, y) just af ter the class 0 packet is admitted is (for x > 0) Z
y 0
f (x, y1 )b0 (y − y1 )dy1 + g(x)b0 (y) /(1 − p(0)).
Let this workload be (x1 , y1 ). Also let the workload of class 0 excluding the service time of the class 0 packet just admitted from y1 be y 0. Let τ (x1 , y1 ) be the first time there is 21
no class 1 work in the system and let at that time y 1 be the class 0 work in the system. We want to know the joint distribution of (τ (x1 , y1), y 1 ) because y 1 will determine when the next packet of class 1 will be admitted to the system (say at time τ (x1 , y1 ) + I1 ) and let at that time the workload in the system be (x2 , y2) (including the work brought in by the arriving class 1 packet). This way we define (xn , yn ), y n and In . We will describe later the method to compute the joint distribution of (τ (xn , yn ), y n ). This allows us to compute the joint distribution of (τ (x1 , y1), I1 , . . . , τ (xn , yn ), In ) for n = 1, 2, . . .. Now define N = inf{n : I1 + I2 + . . . + In > y 0}. Then, the waiting time of the packet under consideration is N X i=1
τ (xi , yi ) + y 0.
This allows us to compute the distribution of the waiting time of the packet but unlike in Section 3, the actual computation of it will be substantially more involved. If we replace in the above computations y 0 by y1 we obtain the sojourn time of the class 0 packet. If x = 0 then the waiting time of the packet of class 0 is obtained in the same way except that now τ (x1 , y1 ) = 0, y1 = y1 . The joint distribution of (τ (x1 , y1 ), y1 ) is obtained by solving the following time dependent rate equation which results from the fact that till time τ (x1 , y1), Vt (1) > 0 and Vt (0) > 0. Also in this equation ft is the pdf of (Vt (1), Vt (0)) starting with f0 the density seen by an admitted class 0 packet (with the additional condition that V0 (1) > 0 because otherwise this equation is not needed) and T > 0 is any time. Z 0
x
Z 0
y
(fT (x0 , y 0 ) − f0 (x0 , y 0 ))dx0 dy 0 Z
=
x
Z
0
−λ1 −λ0
y
0 Z
Z0 0
0
0
0
0
f0 (x , y )dx dy + T
Z
T
Z0 0
x
Z
y
x
Z0
y
0
0
Z 0
T
Z
y 0
(ft (x, y 0 ) − ft (0, y 0))dy 0dt
0
ft (x , y )P (x − x0 < X(1) ≤ s − x0 − y 0 )dx0 dy 0dt ft (x0 , y 0)P (y − y 0 < X(0) ≤ s − x0 − y 0)dx0 dy 0dt.
This system of equations can be solved by discretizing the variables T, x, y as in Section 2.1.3. Then the probability that τ (x1 , y1 ) = t and y1 = y is given by ft (0, y). Actually a similar rate conservation equation can also be written to compute the distribution of I1 by putting the constraint that Vt (1) = 0.
22
4.2
MMPP arrivals
Let {Yt } be the modulating Markov chain with a finite state space. The notations and assumptions here will be the same as in Section 3.2. The process {(Vt (1), Vt (0), Yt )} is a Markov process. The stability, rates of convergence and functional limit theorems for this process as well as for the waiting time and the queue length process hold as for the single class case in the way explained before. Now we compute the stationary distributions for these processes along with the probability of packet loss for each class. The general approach is the same as in Section 3.2. By writing the rate equations as for the Poisson case (with an extra term), one solves them as in Section 3.2. Thus we obtain the stationary probabilities fi (x, y), gi(x), hi (y). To obtain the probability of packet loss and the waiting time distributions for each class, we need the stationary probabilities at the arrival instants of each class. Then the stationary density of (Vt (1), Vt (0), Yt ) to be (x, y, i) at the arrival instant of a class j packet is, as shown in Section 3.2, (for x > 0, y > 0) λi,j fi (x, y) . P k πk λk,j In this formula, fi (x, y) is replaced by gi (x) or hi (y) if V (0) = 0 or V (1) = 0. Thus, the probability of packet loss of a class j, j = 1, 0, packet is X Zi s 0
λi,j P k λk,j πk
Z 0
s
Z 0
s
fi (x, y)(1 − Bi,j (s − x − y))dxdy+
gi (x)(1 − Bi,j (s − x))dx +
Z 0
s
hi (y)(1 − Bi,j (s − y))dy + Pi (1 − Bi,j (s)) . (16)
Next we obtain the waiting time distribution for class 1. The probability that a packet of class 1 is admitted and experiences a delay of x > 0 is Z s−x X 1 λi (1) fi (x, y)Bi,1(s − x − y)dy + gi (x)Bi,1 (s − x) /(1 − p(1)). P 0 k λk (1)πk i
The waiting time distribution of class 0 is more difficult to obtain. However, it can be obtained by the methods used previously. We know the stationary distribution (x1 , y1, i1 ) of the system state as seen by an admitted class 0 packet. To compute the joint distribution of τ (x1 , y1 , i1 ), the class 0 workload and the state of {Yt } at that time, we can write the rate equations for the transient probabilities for this system, as done in Section 3.2. Then the rest of the procedure remains as before.
23
Acknowledgments This work was performed during the stay of Vinod Sharma as a Visiting Professor at the Helsinki University of Technology.
References 1. S. Asmussen, Applied Probability and Queues, Wiley, N.Y., 1987. 2. F. Baccelli and P. Bremaud, Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences, Springer, N.Y., 1994. 3. S. Blake, D. Black, M. Carlson, E. Davies, Z. Wang, W. Weiss, An architecture for differentiated services, Internet RFC 2475, 1998. 4. R. Braden, D. Clark, S. Shenker, Integrated services in the Internet architecture: An overview, Internet RFC 1633, 1994. 5. R.C. Cooper, Introduction to Queueing Theory, 2nd Ed., North Holland, N.Y., 1981. 6. P. Ferguson and G. Huston, Quality of Service: Delivering QoS on the Internet and in Corporate Networks, Wiley, N.Y., 1998. 7. N.K. Jaiswal, Priority Queues, Academic Press, N.Y., 1968. 8. V.V. Kalashnikov, Mathematical Methods in Queueing Theory, Kluwer, Dordrecht, 1994. 9. K. Kawahara, K. Kitajima, T. Takine, Y. Oie, Packet loss performance of selective cell discard schemes in ATM networks, IEEE JSAC, Vol. 15, 1997, 903-913. 10. L. Kleinrock, Queueing Systems, Vol. 1, Wiley, N.Y., 1975. 11. A.Y.-M. Lin and J.Silvester, Priority strategies and buffer allocation protocols for traffic control at an ATM integrated broadband switching system, IEEE JSAC, Vol. 9, 1991, 1524-1536. 12. M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, N.Y., 1989.
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13. V. Ramaswami, Matrix analytic methods: A tutorial overview with some extensions and new results, in Matrix Analytic Methods in Stochastic Models, S.R.Chakravarthy and A.S.Alfa (Ed.), Marcel Dekker, N.Y., 1997. 14. J. Roberts, U. Mocci, J. Virtamo (eds.), Broadband Network Teletraffic – Final Report of Action, COST 242, Springer Lect. Notes in Computer Science, 1996. 15. V. Sharma, Reliable estimation via simulation, Queueing Systems, Vol. 19, 1995, 169192. 16. V. Sharma and N.D. Gangadhar, Discrete time queues with time and space priorities, to appear in 3rd IFIP TC6 workshop on Traffic Management and Design of ATM Networks, London, 1999. 17. Y. Takahashi and O. Hashida, Delay analysis of discrete time priority queue with structured inputs, Queueing Systems, Vol. 8, 1991, 149-164. 18. X. Tan and C. Knessl, Integrated representations and asymptotics for infinite and finite capacity queue described by the unfinished work I, II, SIAM J. Applied Math., Vol. 57, 1997, 791-823, 824-870. 19. W. Wagner, Analysis of a multiserver model with nonpreemptive priorities and nonrenewal input, in J. Labetoulle and J.W. Roberts (ed.) Proc. ITC-14, 1995, 757-767.
25
Laboratory of Telecommunication Technology, Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Finland. Abstract We consider a queue with finite buffer where the buffer size limits the amount of work that can be stored in the queue. The arrival process is a Poisson or a Markov modulated Poisson process. The service times (packet lengths) are iid with a general distribution. Our queue models the systems in the Internet more realistically than the usual M/GI/1/K queue which restricts the number of packets in the buffer rather than the buffer content (the number of bits). We obtain the stability, the rates of convergence to the stationary distribution and functional limit theorems for this system. In addition we also obtain algorithms to compute the stationary density of the workload process, the waiting times and the probability of packet loss. Next we study the queue with two priority classes. The higher priority traffic has preemptive-resume priority. For sharing the buffer we consider two cases. In the first case the buffer is shared by both the classes without any priority. In the second case the buffer is partitioned into two groups one reserved for each class. For this system also we obtain all the results mentioned for the single class traffic. Keywords: Finite buffer queue; MMPP arrival process; Priority classes
1
Introduction
In integrated services networks different QoS can be provided by specifying different space and time priority to the customers. For example, by providing higher time priority to a customer one can lower its time delays. But by providing higher space priority to another customer, we can lower its probability of loss even though, lower time priority means that the delays of these packets can be longer. Thus different customers can be provided differentiated services in a scalable manner. Therefore these priority schemes are being considered in the Integrated Services as well as Differentiated Services Architectures in the Internet (see Braden et al. [4], Blake et al. [3] and Ferguson and Hudson [6]). In this paper we consider a queue with a finite buffer and two priorities of traffic. In contrast to other studies, by a finite buffer of size s we mean that at most s amount of work (e.g. bits) can be stored in the queue at any time. Thus, if a packet of size x arrives but the 1
space available in the buffer is less than x then that packet will be lost. Usually, by a finite buffer one means that at most (say) K packets can be stored in the buffer. If the packet size is fixed (e.g. in ATM networks) then the two notions of buffer size are (almost) the same; otherwise they are not. Thus in the Internet our notion of buffer size is closer to reality. Since the results for our finite buffer queue even with a single class of traffic do not seem to be available, (see however Tan and Knessel [18] for the analysis of the unfinished work for a single class, Poisson arrivals and exponential service times) we will first provide the results for this case. We will compare this queue with the usual M/GI/1/K queue and show that the behaviour of the two queues can be substantially different and hence one should also study the queue with our notion of a buffer length. Interestingly, the traditional method of studying an M/GI/1/K queue at the departure epochs (see e.g., Cooper [5], Chapter 5.9) does not provide a simple way to analyze our queue. This is because the probability of admission of a new arrival depends upon the actual packet lengths of the packets stored at that moment and not just the queue length. However we show that rate conservation equations for the workload process do provide a simple feasible way for this queue. In fact, the workload process is a continuous state space, continuous time Markov process even in the case of a finite buffer queue. Our equation is a generalization of the Tak´acs integrodifferential equation. We will show that our method is effective in solving even the two priority problem. We will also use this method to obtain the results when the arrival process is generalized to a Poisson process modulated by a Markov chain (MMPP). For the two priority system, we will consider two cases. In the first case one of the classes has higher time priority (preemptive resume) but the buffer is completely shared. In the second case, the buffer is completely partitioned into s1 and s0 sizes with s = s0 + s1 although one of the classes has higher time priority. Both of these buffer sharing schemes have been considered in the literature for the usual finite buffer queues. Also, preemptive-resume time priority is a common priority scheme and is being considered in the PPP multilink fragmentation protocol for Internet by IETF. For all the systems considered (with one as well as the two class traffic) we obtain the stability, the rates of convergence and the algorithms to compute the stationary probability of loss of each class, the waiting time distributions and the busy period distribution. Now we mention the other relevant literature. The extensive study of Tan and Knessel [18] was already cited. Infinite buffer priority queues with Poisson arrivals are considered in detail in Jaiswal [7]. A single class queue with Markovian arrivals and infinite buffer is widely studied (see Neuts [12] and Ramaswami [13] for various results and references). Lin and Silvester [11] consider a two class discrete queue with iid traffic and finite buffers. They 2
obtain the probability of packet loss for both the priority classes for various buffer sharing schemes. Some other studies for discrete priority queues are Kawahara et al. [9], Takahashi and Hashida [17] and Sharma and Gangadhar [16]. Wagner [19] considers a continuous queue with Markov modulated arrivals and infinite buffers and obtains the moment generating function of the waiting times for both the priority classes. The delays for a f inite buffer queue with priority and iid and Markov modulated arrivals are considered only in Sharma and Gangadhar [16]. However the methods used in this paper are necessarily very different. The paper is organized as follows. In Section 2 we study the system with a single class of traffic. In Section 3 we will study the two class problem where the two classes have separate buffers to store their packets. The system with a single buffer shared by both the classes is studied in Section 4. In each section we will provide specific algorithms for the stationary waiting times of admitted packets and the probability of packet loss of each class. Some computational results will also be presented and all the results will be generalized to the MMPP arrival traffic.
2
Single class traffic
In Section 2.1 we consider a finite queue with a Poisson arrival process. In Section 2.2 we will generalize our results to the MMPP case.
2.1
Poisson arrivals
We consider the M/GI/1/s-queue with a finite buffer of size s. Let λ be the Poisson arrival rate and the packet sizes be iid with a generic packet length X. The server serves at a unit rate. Also let the total workload (the unfinished work) at time t be Vt . The buffer space is assumed to be allocated dynamically, i.e. when a packet of size X arrives, the packet is admitted into the buffer whenever the total free space in the buffer (= s − Vt ) is greater than or equal to X. In this case, the finite buffer size does not set a constraint on the number of packets but on the total workload (e.g. bits) in the buffer. Since this queueing system has not been studied much before, first we study its tability and the rates of convergence to the stationary distribution in Section 2.1.1. In Section 2.1.2 we will provide the algorithms for the distributions of the workload process, the waiting times and the probability of packet loss. In Section 2.1.3 we will study the busy period.
3
2.1.1
Stability
The process {Vt } is Markov with state space [0, s]. It is also regenerative where we can (and will) take the regeneration epochs as the arrival epochs that find the system empty. Let t = 0 be the arrival epoch of the first customer arriving to the system which finds the system empty and let τ be the number of the next arrival finding the system empty. Since Vt ≤ s at all times, the probability that an arrival will find the system empty is lower bounded by ∆
p1 = P [interarrival time > s] = e−sλ > 0. Thus P [τ > n] ≤ (1 − p1 )n and hence τ has all finite moments and also finite exponential moments in a neighbourhood around zero. If {an } are interarrival times and τˆ is a regeneration length for {Vt } then P τˆ is distributed as τk=1 ak and hence τˆ also has all finite moments and finite exponential moments in a neighbourhood of zero. Furthermore, τˆ has a spread-out distribution. Thus by regenerative process theory (see Asmussen [1], Chapter V) {Vt } has a unique stationary distribution π and the process converges to it in total variation starting from any initial conditions. Also, E[ˆ τ α ] < ∞ for any α ≥ 1 implies (see Kalashnikov [8]) that supA |P [Vt ∈ A] − π(A)| < ct1−α for some constant c. Similarly, from finite exponential moments, we obtain exponential rates of convergence. Furthermore Vt ≤ s and E[ˆ τ α ] < ∞ also imply hR α i τˆ that E < ∞ for any α ≥ 0. Now using Sharma [15], Asmussen [1], we obtain 0 Vt dt functional CLT and LIL theorems for {Vt }. Next we consider other processes of interest. Let Wn be the waiting time of the nth customer. The arrival epochs finding the queue empty are regeneration epochs for {Wn } also with the regeneration length τ . Since P [τ = 1] ≥ p1 , the distribution of τ is aperiodic. Thus, again from regenerative theory, {Wn } has a unique stationary distribution (say π) and sup |P [Wn ∈ A] − π(A)| < C1 n1−α A
for some constant C1 < ∞ and any α > 0. The functional limit theorems hold in the same way. Now consider the queue length process {Lt } where Lt is the number of customers in the system at time t. This process is not necessarily a finite state process. Again τˆ is the regeneration length for {Lt } and hence uniqueness and existence of stationary distributions and rates of convergence to π hold. To show the finiteness of moments of Lt , observe that during a regeneration cycle Lt is upper bounded by the number of customers served in that cycle and hence we have Eπ [Lαt ] < ∞ for all α. 4
In the next subsection we will consider the arrival process to be MMPP. Then we will provide the necessary changes in the arguments to extend these results to the MMPP case. In Section 4 we will be considering the queue with two classes of traffic. Then we will be considering the processes {Vt (i)}, {Wn (i)}, {Lt (i)} for class i, i = 0, 1. Again τ will denote the regeneration length for {(Wn (1), Wn (0))} (any arrival epoch finding the system empty will be a regeneration epoch if we probabilistically assign the class which we always can for our model). These regeneration epochs will remain the same if we consider a single class system with arrival rates λ1 + λ0 and the service distribution as the mixture of the service distributions of the two classes (with probabilities λ1 /(λ1 + λ0 ) and λ0 /(λ1 + λ0 )). Thus, from the single class result we also obtain E[τ α ] < ∞ and E[ˆ τ α ] < ∞ for the two class problem and all the above results carry over directly. The system in Section 3 requires a slight modification to the above arguments. We will mention them there.
2.1.2
Workload distribution
Our task in this subsection is to determine the stationary distribution F of the workload (unfinished work) V in the buffer, the packet loss probability p, the traffic loss probability p(t) (the fraction of number of bits lost) and the waiting time distribution. Due to the PASTA property, p = P{X + V > s}. Note that p and p(t) are two different quantities, because the loss process depends on the packet size. First we establish the existence of density f (x) of F (x) for x > 0. Let A be a Borel subset of IR such that 0 6∈ A and Lebesgue measure of A is 0. From the dynamics of the queue, R R for any initial distribution, 0∞ 1{Vt ∈A} dt = 0 a.s. Therefore, π(A) = Eπ [ 0τ 1{Vt ∈A} dt]/E[τ ] = R
R
Eπ [ 0∞ 1{Vt ∈A} 1{τ >t} dt]/E[τ ] ≤ Eπ [ 0∞ 1{Vt ∈A} dt]/E[τ ] = 0. Thus F is absolutely continuous with respect to Lebesgue measure on {x : x > 0}.
We use a Tak´acs type approach to compute F . Let B(x) and b(x) be the cdf and the pdf of the packet size X. Consider the balance of probability flows across a control surface at the workload level x under stationarity. The downward flow, due to the emptying of the buffer, is equal to the probability density at x, f (x) = F 0 (x). The upward flow is due to arrivals. An arrival causes the system state to cross the control level at x provided that the buffer content before the arrival, denoted by y, is less than or equal to x and that the size of the arriving packet is greater than x − y (in order to cross the control level) but less than
5
s − y (in order for the packet to be admitted into the buffer). Thus we obtain
Z
f (x) = λ P0 Q(x, 0) +
0
x
Q(x, y)f (y)dy ,
(1)
where Q(x, y) = B(s − y) − B(x − y) and P0 is the (so far unknown) probability of an empty queue. In this form, written in terms of the pdf function f (x), the Tak´acs equation is an integral equation, a Volterra equation of the second kind (in contrast to an integro-differential equation; see Kleinrock [10], Chapter 5.12, pp. 226-230). We note from the equation that the value of P0 is just a scaling factor, i.e. the solution to this equation is proportional to P0 . Then, it is sufficient that we solve the equation for the case P0 = 1. From now on we let φ(x) denote the solution corresponding to this case; the solution for any value P0 is then f (x) = P0 φ(x). Let us further denote Z
I=
s
0
φ(x)dx.
In order to get a distribution with unit probability we must have P0 + P0 I = 1
(2)
from which
1 . 1+I With P0 known, the distribution of the unfinished work is completely determined. It consists of an atom P0 at zero and of a continuous distribution with the pdf P0 φ(x) in the interval (0, s). P0 =
Let us take the mean packet size m = E[X] as the unit of work and denote by ρ = λm the (offered) load of the system. The admitted load ρ0 is equal to 1 − P0 . Thus the fraction of traffic lost is p(t) = (ρ − ρ0 )/ρ = 1 −
I/ρ . 1+I
(3)
The packet loss probability is
p = P0 1 − B(s) +
Z 0
s
(1 − B(s − y))φ(y)dy .
(4)
The Volterra equation (1), with P0 = 1, can be solved in a straightforward way starting from the value at x = 0 and proceeding to greater values. First, note that f (0) is directly determined by the equation, f (0) = λ Q(0, 0) = λ B(s). 6
Second, for any x the value of f (x) is determined by the values of f (y) for y ≤ x. In particular, consider a numerical scheme, where we wish to determine the values of f (x) at the grid points xi = i∆ for i = 0, . . . , n with s = n∆. Consider now Eq. (1) at xi . Whatever numerical scheme is used for the evaluation for the integral, the value of the integral is a linear combination of the values of f (xj ) for j = 0, . . . , i. Thus, (1) is a linear equation for f (xi ) which can be solved in terms of the previous values of f (xj ), j = 0, . . . , i − 1. So starting from i = 0 we can solve the values consecutively for greater values of i up to n. We have solved the Volterra equation as described above, using Simpson’s rule for the integration (with a slight modification for an odd number of intervals), and calculated the traffic and packet loss probabilities p(t) and p from equations (3) and (4) for an M/M/1/s system with the buffer size s = 3 (in units of mean packet size). The loss probabilities are plotted as a function of the offered load ρ in Fig. 1. For comparison we have also plotted the curve for an ordinary M/M/1/K system with K = 3 buffer places (including the buffer space of the server). In this case the packet and traffic loss probabilities are equal and are given by ρK /(1 + ρ + · · ·+ ρK ). Of course as s → ∞ the stationary distribution of each queue tends to that of M/N/1 queue. This assertion holds for the general service time distribution also. 0.6
loss probability
0.5 traffic loss 0.4
loss with 3 buffer places
0.3 0.2
packet loss 0.1 0 0.25
0.5
0.75
1
1.25
1.5
1.75
2
offered load
Figure 1: Packet and traffic losses for a finite queue with exponential packet size distribution. Observe the rather high level of p(t) for low loads. When the load approaches zero, each arriving packet finds the system empty. Then the probability that a packet is not admitted is P [X > s] = e−s . The average size of a lost packet E[X|X > s] is (s + 1) resulting in the traffic loss probability p(t) ≈ E[X|X > s]P [X > s] = (s + 1)e−s . For s = 3 the value is 7
numerically around 4 · 0.05 = 0.2 as shown in the figure. The finite loss probabilities at low loads in the above example (as well as in some later examples) are due to the assumed exponential packet size distribution. It would be more realistic to assume the packet size to be bounded. This poses no problem for the numerical solution, and we demonstrate this by another example shown in Fig. 2. Here the packet size distribution is Exp(1) truncated at 1, with mean 0.418. The buffer size is 1.254, i.e. three times the mean. In this case the blocking probabilities tend to zero at low loads, since the buffer can always hold at least one packet. 0.6 loss with 3 buffer places
loss probability
0.5 0.4 0.3 traffic loss 0.2
packet loss 0.1 0 0.25
0.5
0.75
1
1.25
1.5
1.75
2
offered load
Figure 2: Packet and traffic losses for a finite queue with truncated exponential packet size distribution. By PASTA, under stationarity the workload seen by an arriving customer also has the cdf F and the pdf f . Then the density of the stationary waiting time of an admitted packet is (for x > 0) P [V ∈ (x, x + dx)|a packet is admitted] =
P [a packet is admitted|Vt ∈ (x, x + dx)]f (x)dx P [a packet is admitted]
= f (x)B(s − x)dx/(1 − p). Also the probability that an admitted packet does not wait is P0 B(s)/(1 − p). Similarly, the sojourn time of an admitted packet has density Z 0
x
f (y)b(x − y)dy/(1 − p) + P0 b(x)/(1 − p).
8
2.1.3
Busy periods
We will need the distribution of the busy periods of the queue in later sections for calculation of the waiting times of the lower priority packets. Therefore, we compute these quantities now. We denote by Tb (x) the busy period if the period starts (say at time t = 0) with initial workload of x. We are not assuming that there is an arrival at t = 0 and hence Tb (0) = 0. Let S(x) be the work that enters the queue during time [0, x]. We will calculate the distribution of S(x) later on. Then, we can solve the following equation satisfied by E[Tb (x)] Z
E[Tb (x)] = x +
s 0
E[Tb (y)]dP [S(x) = y].
(5)
To solve this equation numerically we discretize x to x0 = 0 < x1 < · · · < xn = s. Then for E[Tb (xi )], i = 0, . . . , n, we get a system of (n + 1) linear equations in (n + 1) variables. Observing that E[Tb (0)] = 0, we can then solve the other n unknowns from the resulting n equations. Similarly for E[Tb (x)2 ] we get the integral equation E[Tb (x)2 ] =
Z
x
0
E[(x + Tb (y))2]dP [S(x) = y]
= x2 + 2x
Z
0
s
Z
E[Tb (y)]dP [S(x) = y] +
s
0
E[Tb (y)2]dP [S(x) = y].
(6)
Now from the distribution of S(x) and the solution of (5), we can solve for (6). Higher moments of Tb (x) can be obtained in the same way. Furthermore, the density fx of Tb (x) satisfies the equation
Z
fx (y) =
s
0
fz (y − x)dP [S(x) = z]
where y ≥ x. This equation can be solved by discretizing the variables x, y and z. If T0 denotes a busy period started by an arriving packet, its moments (and distribution) can be obtained from that of Tb (x) using the relation E[T0α ]
Z
=
0
s
E[Tb (x)α ]b(x)dx.
Now we obtain the distribution of S(x) using the rate equations for transient probabilities. We have, for h > 0, P (Vt+h ≤ y) = P (Vt ≤ y + h)(1 − hλ) + hλ
Z 0
y
ft (z)[B(y − z) + (1 − B(s − z))]dz + o(h)
where ft is the pdf of Vt given V0 = x. Subtract P (Vt ≤ y), divide by h and let h go to zero. Using P (Vt = 0) = 0 for 0 ≤ t < x we get by integrating over time from 0 to T ≤ x Z 0
y
(fT (z) − f0 (z))dz =
Z 0
T
(ft (y)dt − ft (0))dt − λ 9
Z 0
T
Z 0
y
ft (z)Q(y, z)dzdt.
By discretizing for y, z and t we can solve this equation and then fx provides the density of S(x). Observe that there is one equation for each pair of discretized values of y and T . Thus there are as many equations as there are unknowns variables ft (y) at the discretized points. Using similar rate equations for transient probabilities, we can also compute the distribution of Tb (x).
2.2
MMPP arrivals
Consider the case where the input process is MMPP with an n-state modulating process {Yt }. Let {Yt } have the transition rate matrix with elements qi,j . Let λi denote the Poisson arrival rate and Bi (t) the distribution of the arriving packet lengths when {Yt } is in state i. Now we obtain the existence, uniqueness, rates of convergence etc. results for this system. Define λ = max λi . Then, whenever an arrival comes with an interarrival time greater than s (this happens with probability lower bounded by e−λs > 0), the system is empty. Thus, if τ denotes the number of the first arrival (if the system starts at t = 0 with an arrival to an empty system) to the empty system, we obtain, as in the last subsection, E[τ α ] < ∞ for all α ≥ 0. However, these epochs are not the regeneration epochs for this system. Now consider a state i0 for {Yt } which is assumed to be reachable from any other state. We will consider an arrival epoch to the empty system when Yt = i0 as a regeneration epoch. Then, because of finite state space there is a 0 < ∆ < s and a δ > 0 such that P [Y∆ = i0 |Y0 = i] > δ > 0 for all i. Hence, if an arrival comes to an empty system then a time s after it, with probability lower bounded by δe−(qi0 ,i0 +λ)s the system is empty and Yt is in state i0 . Therefore, with probability lower bounded by δe−(qi0 ,i0 +λ)s e−qi0 ,i0 δ1 (1 − e−λi0 δ1 ) within time s + δ1 , an arrival comes to the empty system and {Yt } is in state i0 . Therefore denoting by τ˜ the regeneration length (in terms of number of arrivals) and τˆ the regeneration length of the process {(Vt , Yt )}, we obtain E[˜ τ α ] < ∞ and E[ˆ τ α ] < ∞ for any α > 0. Finiteness of exponential moments in a neighbourhood of 0 can also be shown. Thus we obtain uniqueness, existence of the stationary distribution of {(Vt , Yt )} as well as rates of convergence and function limit theorems. Similarly we can obtain the results for the queue length and the waiting time process. Next we obtain algorithms to compute the stationary density of (Vt , Yt ), the waiting time distributions and the probability of packet loss.
10
Define Fi (x) = P [V ≤ x, Yt = i] and let fi (x) be the density of Fi (x) for x > 0. In the same way as before we can write the balance equation for the stationary system. Consider the set of states {V ≤ x, Yt = i} and write the balance of probability flows across a control surface enclosing this set of states. This gives Z Fi0 (x) +
X j
qj,iFj (x) = λi
x
0
Qi (x, y)dFi(y).
where Qi (x, y) = Bi (s − y) − Bi (x − y). In terms of the pdf’s, extracting the atoms at 0, this becomes
Z
fi (x) = λi Pi Qi (x, 0) +
x
0
Qi (x, y)fi(y)dy −
X j
qj,i Pj +
Z 0
x
fj (y)dy ,
(7)
where Pi = Fi (0). Now Eq. (7) can be solved in the same way as before, proceeding from 0 to greater values of x: fi (x) depends on the values of fj (y), j = 1, . . . , n for y ≤ x only. The essential difference is that now we have several unknowns, Pi , i = 1, . . . , n. Suitable boundary conditions have to be formulated in order to solve these values. Since the buffer content is always ≤ s we have Fi (s) = πi , where the πi are the steady state probabilities of Yt . Denote Z s Z s I = ( f1 (x)dx, . . . , fn (x)dx). 0
0
Now we can solve Eq. (7) for the specific initial condition Pi = 1 and Pj = 0 for j 6= i. Let Ii be the value of the vector I resulting from these initial conditions. Then the Pi can be solved from the equation P+
X i
Pi Ii = π,
where P = (P1 , . . . , Pn ). This generalizes the normalization condition (2) for the Poisson input case. Next we consider the stationary probability of packet loss and the waiting times of the admitted packets. For this we need to compute the stationary probability of {Vt } seen by an arriving packet. This is the Palm probability corresponding to the time stationary probability of {(Vt , Yt )} computed above and hence can be computed via the general formulae (see e.g. Baccelli and Bremaud [2]). However, for the MMPP case we can actually have a simple direct formula which we now provide. Let N(a, b) be the number of arrivals under stationarity to the system during the time interval (a, b). Then lim P [Vt ≤ x, Yt = i|N(t, t + h) = 1] h↓0
11
P [N(t, t + h) = 1|Vt ≤ x, Yt = i]P [Vt ≤ x, Yt = i] h↓0 P [N(t, t + h) = 1] λi Fi (x) . = P j πj λj
= lim
Thus the probability of packet loss p is p=
X
P
i
λi [Pi (1 − Bi (s)) + j πj λj
Z
s
0
fi (x)(1 − Bi (s − x))dx]
and the probability that an admitted packet has to wait time x > 0 (similarly one can write for x = 0) has the density X λi fi (x) Bi (s − x) . P j πj λj (1 − p) i Of course now the sojourn time distribution can also be computed. We illustrate the computation of the probability of packet loss, the fraction of traffic lost P (= 1 − (1 − Pi )/ρ) and the pdf of the stationary waiting times via the above equations by an example. Consider a queue with {Yt } having two states with transition rates q1,2 = 0.1, q2,1 = 0.3, λ1 = 0.5, λ2 = 2.5 and service times exponentially distributed with mean 1. The probability of packet loss is shown in Fig. 3 and the waiting time density in Fig. 4 (it has an additional atom at 0 of mass 0.789). 0.6
loss probability
0.5 traffic loss 0.4 0.3 packet loss 0.2 0.1 0 0.25
0.5
0.75
1
1.25
1.5
1.75
2
offered load
Figure 3: Probability of packet and traffic loss in MMPP queue. To compute the waiting time distribution of the lower priority packets in Section 3.2, we will need the distribution of Tb (i, x), the first time the queue becomes empty if started at t = 0 in state (i, x). The moments and distribution of Tb (i, x) can be computed as for the Poisson case with slight changes. We show it by calculating the joint distribution of Tb (i, x) 12
0.25
0.2 0.15 0.1 0.05 0 0.5
1
1.5
2
2.5
3
waiting time Figure 4: Waiting time density for MMPP queue. and YTb (i,x) which is obtained by solving the following simultaneous equations XZ
P [Tb (i, x) = y, YTb(i,x) = j] =
i1
P [Vx ∈ (x1 , x1 + dx), Yx = i1 |V0 = x, Y0 = i]
·P [Tb(i1 , x1 ) = y − x, Yy−x = j]. By discretizing the variables x, y we can solve these equations if we also know P [Vx ∈ (x1 , x1 + dx), Yx = i1 , |V0 = x, Y0 = i]. One way to obtain this transition matrix is by solving the rate equations corresponding to the transient probabilities. To obtain these, observe that for h > 0 small, P [Vt+h ≤ y, Yt+h = j] = (1 − hλ Z j − hqj,j )P [Vt ≤ y + h, Yt = j] y
+hλj +
X
k6=j
0
P [Vt = z, Yt = j](Bj (y − z) + (1 − Bj (s − z))dz
hqk,j P [Vt ≤ y + h, Yt = k] + o(h).
Now subtract P (Vt ≤ y, Yt = j) from both sides, divide by h and let h tend to 0. Since the state Vt = 0 will not be visited till time Tb (i, x), we obtain, for T ≤ x Z
y 0
(fT,j (z) − f0,j (z))dz =
Z 0
+
T
ft,j (y)dt − λj
X k
Z
qk,j
0
T
Z 0
y
Z 0
T
Z 0
y
ft,j (y 0 )Qj (y, y 0)dy 0
ft,k (y 0)dy 0dt.
This system of equations can be solved as in the last section except that we have to solve them simultaneously.
13
3
Two priority classes with completely partitioned buffer
3.1
Poisson arrivals
In this section we consider a system with two classes. Class 1 traffic has preemptive-resume priority over class 0 traffic, i.e. whenever there is no class 1 traffic in the system the remaining service of class 0 packets resumes. There is a separate buffer for each class. The arrival process of each class is Poisson and the service times are iid with general distributions. For class i the arrival rate is λi , a generic packet length is X(i) with cdf Bi and pdf bi and its buffer can store a total work of si at a time. Let Vt (i) be the buffer content at time t in the buffer of class i. The existence and uniqueness of stationary distributions of (Vt (1), Vt (0)), the waiting times and the queue length process follow in the same way as in Section 2. Similarly, the rates of convergence hold. Indeed observe that we can form this system by considering a single Poisson stream with arrival rate λ1 + λ0 . On arrival a packet is made of class 1 with probability λ1 /(λ1 + λ0 ); otherwise of class 0. Then any arrival epoch finding the system empty will be a regeneration epoch. Since this can happen at any arrival time with probability lower bounded by e−(s1 +s0 )(λ1 +λ0 ) we can now repeat the arguments for the single class case. Now we will provide algorithms to compute the density of the stationary distribution of (Vt (1), Vt (0)) via the rate conservation equations. Then we will compute the stationary probability of packet loss, and the waiting time distributions of each class. We will obtain the rate conservation equations in several steps. Also, we will denote the density of P [0 < V (1) ≤ x, V (0) = 0] by g(x), the density of P [V (1) = 0, 0 < V (0) ≤ y] by h(y), the density of P [0 < V (1) ≤ x, 0 < V (0) ≤ y] by f (x, y) and the stationary probability that the system is empty by P0 . The existence of the respective densities can be shown by an argument similar to the one used in subsection 2.1.2 for the one-dimensional single class case. The rate equation at (V (1) = 0, V (0) = 0) is g(0) + h(0) = P0 (λ1 B(s1 ) + λ0 B0 (s0 )).
(8)
Similarly the rate equation for the set (0 < V (1) = x, V (0) = 0) is Z
g(x) + P0 λ1 B1 (x) = g(0) + λ1
0
x
0
0
0
g(x )Q1 (x, x )dx + λ0 B0 (s0 ) 14
Z 0
x
g(x0 )dx0 ,
(9)
where Qi (x, x0 ) = Bi (si − x0 ) − Bi (x − x0 ), i = 0, 1. (This equation is illustrated in Fig. 5 where each arrow corresponds to one term in the equation). By discretization we can solve for g(xi ), xi > 0 in terms of g(0). Finally, we can obtain g(0) by observing that g(s1 ) = 0. Next consider the rate equation for the set (V (1) = 0, 0 < V (0) ≤ y) Z
h(y) + P0 λ0 B0 (y) +
0
y
f (0, y 0)dy 0 Z
= h(0) + λ1 B1 (s1 )
y
0
Z
h(y 0)dy 0 + λ0
y
h(y 0 )Q0 (y, y 0)dy 0.
0
(10)
This equation can again be solved for h(yi ) by discretization for successively larger values yi if we also know h(0) and f (0, y 0), 0 < y 0 < y. The initial value h(0) can be obtained from (8) by plugging in g(0) obtained above (it is still in terms of the unknown P0 to be found later on). To get f (0, y 0) we need the rate equations for the strip (0 < V (1) ≤ x, V (0) ∈ (y, y + dy)), which are Z
f (x, y) + h(y)λ1 B1 (x) + λ0 b0 (y) Z
= f (0, y) + λ1
x
0
x
0
0
Z
0
g(x )dx + λ0
x
0
0
dx
f (x0 , y)Q1(x, x0 )dx0 + λ0 B0 (s0 − y)
Z
0
Z
x 0
y
f (x0 , y 0)b0 (y − y 0 )dy 0
f (x0 , y)dx0.
(11)
We illustrate this equation in Fig. 5. Finally we obtain g(xi), h(yi ), f (xi , yi) in terms of P0 which is then obtained from the normalization equation Z
P0 +
0
s1
Z
g(x)dx +
s0
0
Z
h(y)dy +
0
s1
Z 0
s0
f (x, y)dxdy = 1.
These equations can be solved by discretizing x and y variables by keeping y fixed for various values of x if we know h(y) and f (0, y) and f (x0 , y 0), y 0 < y. We will obtain f (0, y) by fixing f (s1 , y) = 0. Then f (0, y) can be used to obtain h(y 00 ) for y 00 > y from Eq. (10) which again will be used in (11) to obtain f (x, y 00 ) and so on. The initial conditions of f (x0 , y1) will be obtained from (11) by observing that the integral containing f (x0 , y 0) can then be made zero. The above discretization can be used to approximate f (x, y). The finer the discretization is, the better is the approximation. We use this approximation to obtain other quantities of interest. Consider the class 1 traffic. This traffic is not affected by the class 0 traffic. Therefore, the probability of packet loss p(1) of class 1, and the waiting time distribution are obtained as for the single class traffic in the last section. Now consider the class 0 traffic. By PASTA, on arrival a class 0 packet sees the workload of y in its queue with the probability h(y) +
R s1 0
∆
f (x, y)dx = f0 (y). Then the probability p0 that a class 0 packet is lost is (1 − B0 (s0 ))(P0 +
Z 0
s1
Z
g(x)dx) + 15
0
s0
f0 (y)(1 − B0 (s0 − y))dy.
y
y
s0
s0
s1
x
s1
x
Figure 5: Illustration of the probability rate flows in Eq. (9) (on the left) and Eq. (11) (on the right). Also, the rate of the work entering the system is ρ0 = 1 − P0 . The fraction of class 0 traffic lost is
Z
(P0 +
0
s1
Z
g(x)dx)
∞
s0
Z
zb0 (z)dz +
0
s0
Z
(h(y) + f (x, y))
∞
s0 −y
zb0 (z)dz /E[X(0)].
Similarly one can write the fraction of class 1 traffic lost. We illustrate the computation of the above equations by an example. We take λ0 = 1.5λ1 , the mean service rate 1, and all the distributions are exponential. The two buffer sizes are s0 = 5 and s1 = 3. Fig. 6 provides the probabilities of the packet loss and the traffic loss as a function of λ1 . Fig. 7 gives the plot of f (x, y) and Fig. 8 of g(x) and h(y) when λ1 = 0.2. 0.5
loss probability
0.4 0.3
class 0, traffic loss
class 1, traffic loss class 0, packet loss
0.2 0.1
class 1, packet loss
0 0.2
0.4
0.6
0.8
1
offered load of class 1 traffic
Figure 6: Packet loss probability for the two classes. Next we compute the waiting time distribution of a class 0 packet (till the time its service first starts) that is admitted. By PASTA, an arriving class 0 packet sees the workload 16
5 4 3 2 1 0.05 0.04 0.03 0.02 0.01 0 0 1 2 3
Figure 7: Waiting time density for x > 0, y > 0. (Vt (1) = x, Vt (0) = y) with stationary density f (x, y) (or g(x) or h(y) as the case may be). Thus the stationary density seen by an admitted class 0 packet is f (x, y)B0(s −y)/(1 −p(0)). Now we compute the waiting time of an admitted class 0 packet (denote its arrival time as time 0) conditioned on it seeing the work (x1 , y1 ) on its entrance. The first time the class 1 queue will be empty will be denoted by τ (x1 ) which is independent of y1 and its distribution and moments can be computed as for a single class queue in Section 2. Let the first arrival of class 1 after time τ (x1 ) that is admitted occur at time τ (x1 ) + I1 . I1 is exponentially distributed with parameter λ1 B1 (s1 ). During time [τ (x1 ), τ (x1 ) + I1 ) class 0 packets can be served. If I1 > y1 then the service of class 0 packet under consideration will start during this time and its waiting time will be τ (x1 ) + y1 . Otherwise this packet will have to wait for the next time when class 1 queue is empty. Let that time be τ (x1 ) + I1 + τ1 . Let {τn } be iid with the distribution of τ1 (which is known from Section 2) and let {In } be iid with the distribution of I1 . Let N = inf{n : I1 + I2 + . . . + In > y1 }. Then N − 1 is Poisson distributed with parameter λ1 B1 (s1 )y1 . The waiting time distribution of the packet under consideration is τ (x1 ) +
N −1 X i=1
τi + y1 ,
where τ (x1 ) is independent of {τn }. Thus, we can calculate the distribution (and moments) of the waiting times of the packet. 17
0.2
0.15
0.1
h(y) g(x)
0.05
0 1
2
3
4
5
x,y
Figure 8: Waiting time density for x = 0, or y = 0.
3.2
MMPP arrivals
Consider a finite state Markov chain {Yt } with transition rates qi,j which modulates the arrival process in the following way. If during time [t, t + h], {Yt } is in state i then during this interval class j, j = 0, 1 arrivals occur according to a Poisson process with rate λi,j and the service time distribution of these arrivals have cdf Bi,j and pdf bi,j . Now the process {(Vt (1), Vt (0), Yt )} is Markov. Also, the existence and uniqueness of stationary distributions, rates of convergence and functional limit theorems hold as for the single class case using the additional arguments of Section 3.1. Next we provide the algorithms for computing the stationary density of {(Vt (1), Vt (0), Yt)}, the waiting distributions and the probability of packet loss for each class. First we provide the rate conservation equations for {(Vt (1), Vt (0), Yt )}. We will denote the stationary density at (x, y, i), x > 0, y > 0 by fi (x, y), for x = 0, y > 0, i by hi (y) and for x > 0, y = 0, i by gi (x). The rate equation at state (0, 0, i) is gi(0) + hi (0) +
X j
qj,iPj = Pi
X
λi,k Bi,k (sk ),
k
and on the axis 0 < V (1) ≤ x, V (0) = 0, Y = i is gi(x) + Pi λ1 Bi,1 (x) + Z
= gi (0) + λi,1
0
x
0
X j
Z
qj,i
x
0
gj (x0 )dx0
0
0
gi (x )Qi,1 (x, x )dx + λi,0 Bi,0 (s0 )
Z 0
x
gi (x0 )dx0 ,
where Qi,j (x, x0 ) = Bi,j (sj − x0 ) − Bi,j (x − x0 ). Similarly, on the axis V (1) = 0, 0 < V (0) ≤ 18
y, Y = i we get Z
hi (y) + Pi λi,0 Bi,0 (y) +
y 0
fi (0, y 0)dy 0 + Z
= hi (0) + λi,1 Bi,1 (s1 )
0
y
X j
0
Z
qj,i
y
0
0
hj (y 0)dy 0
Z
hi (y )dy + λi,0
0
y
hi (y 0)Qi,0 (y, y 0)dy 0
and for 0 < V (1) ≤ x, V (0) ∈ (y, y + dy), y > 0 Z
fi (x, y) + hi (y)λi,1Bi,1 (x) + λi,0 bi,0 (y) Z
+λi,0
x
0
dx0
Z
y 0
Z
= fi (0, y) + λi,1
0
0 0
x
gi (x0 )dx0
fi (x0 , y 0 )bi,0 (y − y )dy 0 + x
X j
qj,i fj (x, y)
fi (x0 , y)Qi,1(x, x0 )dx0 + λi,0 Bi,0 (s0 − y)
Z 0
x
fi (x0 , y)dx0.
These equations are solved numerically using previous methods. To obtain the probability of packet loss and the waiting time distributions, we need the stationary probability of {Vt (1) ∈ (x, x + dx), Vt (0) ∈ (y, y + dy) and Yt = i} at an arrival P instant of class j. For x > 0, y > 0 this probability is λi,j fi (x, y)/( k λk,j πk ). Similarly one can obtain the density for x = 0, y > 0; x > 0, y = 0 and the atom at x = y = 0. Therefore, the probability of packet loss of class 1 is Z s1 Z s0 X 1 p(1) = P λi,1 gi (x) + fi (x, y)dy (1 − Bi,1 (s1 − x))dx 0 0 k λk,1 πk i Z
+
s0
0
hi (y)dy + Pi (1 − Bi,1 (s1 )) .
Similarly, one can calculate the probability of packet loss for class 0. The probability that an admitted packet of class 1 experiences a delay of x > 0 (similarly one can write for x = 0) is "
#
Z s0 X 1 1 . λi,1 fi (x, y)dy + gi (x) Bi,1 (s1 − x) P 1 − p(1) 0 k λk,1 πk i
Next we compute the waiting time distribution of an admitted packet of class 0. For this, we need the stationary system state probability as seen by an admitted class 0 packet. This can be calculated as above and the probability that it sees V (1) = x, V (0) = y, Y = i is given by (say for x > 0, y > 0) P
1 Bi,0(s0 − y) . λi,0 fi (x, y) 1 − p(0) k λk,0 πk
Let the state an admitted (say at time t = 0) class 0 packet sees be (x1 , y1, i1 ). Then, the joint distribution of the first time the class 1 packets are not in the system (denoted by 19
τ (x1 , i1 )) and the state i2 of the chain {Yt } at time τ (x1 , i1 ) can be computed as for the single class result provided in Section 2.2. Let τ (x1 , i1 ) + I1 be the time the first packet of class 1 is admitted to the queue after time τ (x1 , i1 ). Given the distribution of i2 , the distribution of I1 can be computed and also the distribution of the system at time τ (x1 , i1 ) + I1 (these can be computed by writing the rate conservation equations for transient probabilities just as at the end of Section 2.2). Now the distribution of the waiting time of the admitted class 0 packet can be computed as for the Poisson case although the actual computations are much more involved.
4 4.1
Queue with completely shared buffer Poisson arrivals
We consider the system of the last section except that now there is only one buffer of length s. The rest of the notation and assumptions are as in the last section. There is no priority for either class for storing in the buffer, i.e. if a packet of either class comes and there is space in the buffer to store the packet, the packet will be admitted. The existence and uniqueness of stationary distributions for this system were already obtained in Section 2.1. There we also obtained the rates of convergence and the functional limit theorems. Next we obtain the stationary probability density of the buffer content (Vt (1), Vt (0)), the probability of loss and the distribution of waiting times of the two classes of packets. We will follow the approach of the last section. First we will obtain the integral equations of f (x, y), g(x) and h(y) via the rate conservation equations. We will solve these equations iteratively by discretization. Then we will calculate the stationary probability of loss and waiting times. This system is more complicated than that of the last section. Now we write the rate conservation equations to compute the stationary densities f (x, y), g(x) and h(x) of (V (1), V (0)). The overall procedure and notations are as in the last section. The rate equation at (0, 0) is g(0) + h(0) = P0 [λ1 B1 (s) + λ0 B0 (s)].
(12)
The rate equation for the set {0 < V (1) ≤ x, V (0) = 0} is Z
g(0) + λ1
0
x
0
0
g(x )Q1 (x, x ) + λ0
Z 0
x
g(x0 )B0 (s − x0 )dx0 = g(x) + P0 λ1 B1 (x), 20
(13)
where Qi (x, x0 ) = Bi (s − x0 ) − Bi (x − x0 ), and for the set {V (1) = 0, 0 < V (0) ≤ y} Z
h(0) + λ1
0
y
0
0
0
h(y )B1 (s − y )dy + λ0 Z
= h(y) + P0 λ0 B0 (y) +
0
y
Z
y
0
h(y 0 )Q0 (y, y 0)dy 0
f (0, y 0)dy.
(14)
Finally, the rate equation for the strip {0 < V (1) ≤ x, V (0) ∈ (y, y + dy)} is Z
f (x, y) + h(y)λ1B1 (x) + λ0 b0 (y) Z
= f (0, y) + λ1
x
0
0
x
g(x0 )dx0 + λ0
Z
f (x0 , y)Q1(x, x0 − y)dx0 + λ0
x
Z0
dx0
x
Z
y 0
dy 0 f (x0 , y 0)b0 (y − y 0)
B0 (s − y − x0 )f (x0 , y)dx0.
0
(15)
We solve Eqs. (12)-(15) via discretization of (x, y) variables to obtain the values for g(x), h(y) and f (x, y) in exactly the same way as in the last section. We compute the probability of packet loss of each class based on the stationary probabilities computed above. By PASTA, on arrival a class i packet sees (Vt (1) = x, Vt (0) = y) work with probability density f (x, y) or g(x) or h(y) as the case may be. Thus, the probability that a class i packet is lost is Z
p(i) =
s
0
+
Z
Z
0 s
0
s
f (x, y)(1 − Bi (s − x − y))dxdy +
Z 0
s
g(x)(1 − Bi (s − x))dx
h(y)(1 − Bi (s − y))dy + P0 (1 − Bi (s)).
Also, the amount of work entering the system in unit time is given by ρ0 = 1 − P0 . By PASTA, the probability density that an admitted class 1 packet will be delayed by time x > 0 is Z 0
s−x
f (x, y)B1(s − x − y)dy + g(x)B1 (s − x) /(1 − p(1)).
Next we calculate the waiting time distribution of the class 0 packets. This is substantially more difficult to compute. Using PASTA, the stationary probability density of (Vt (1), Vt(0)) at (x, y), x > 0, y > 0 as seen by an admitted (say at time 0) class 0 packet is f (x, y)B0(s − x − y)/(1 − p(0)). Thus the probability density at (x, y) just af ter the class 0 packet is admitted is (for x > 0) Z
y 0
f (x, y1 )b0 (y − y1 )dy1 + g(x)b0 (y) /(1 − p(0)).
Let this workload be (x1 , y1 ). Also let the workload of class 0 excluding the service time of the class 0 packet just admitted from y1 be y 0. Let τ (x1 , y1 ) be the first time there is 21
no class 1 work in the system and let at that time y 1 be the class 0 work in the system. We want to know the joint distribution of (τ (x1 , y1), y 1 ) because y 1 will determine when the next packet of class 1 will be admitted to the system (say at time τ (x1 , y1 ) + I1 ) and let at that time the workload in the system be (x2 , y2) (including the work brought in by the arriving class 1 packet). This way we define (xn , yn ), y n and In . We will describe later the method to compute the joint distribution of (τ (xn , yn ), y n ). This allows us to compute the joint distribution of (τ (x1 , y1), I1 , . . . , τ (xn , yn ), In ) for n = 1, 2, . . .. Now define N = inf{n : I1 + I2 + . . . + In > y 0}. Then, the waiting time of the packet under consideration is N X i=1
τ (xi , yi ) + y 0.
This allows us to compute the distribution of the waiting time of the packet but unlike in Section 3, the actual computation of it will be substantially more involved. If we replace in the above computations y 0 by y1 we obtain the sojourn time of the class 0 packet. If x = 0 then the waiting time of the packet of class 0 is obtained in the same way except that now τ (x1 , y1 ) = 0, y1 = y1 . The joint distribution of (τ (x1 , y1 ), y1 ) is obtained by solving the following time dependent rate equation which results from the fact that till time τ (x1 , y1), Vt (1) > 0 and Vt (0) > 0. Also in this equation ft is the pdf of (Vt (1), Vt (0)) starting with f0 the density seen by an admitted class 0 packet (with the additional condition that V0 (1) > 0 because otherwise this equation is not needed) and T > 0 is any time. Z 0
x
Z 0
y
(fT (x0 , y 0 ) − f0 (x0 , y 0 ))dx0 dy 0 Z
=
x
Z
0
−λ1 −λ0
y
0 Z
Z0 0
0
0
0
0
f0 (x , y )dx dy + T
Z
T
Z0 0
x
Z
y
x
Z0
y
0
0
Z 0
T
Z
y 0
(ft (x, y 0 ) − ft (0, y 0))dy 0dt
0
ft (x , y )P (x − x0 < X(1) ≤ s − x0 − y 0 )dx0 dy 0dt ft (x0 , y 0)P (y − y 0 < X(0) ≤ s − x0 − y 0)dx0 dy 0dt.
This system of equations can be solved by discretizing the variables T, x, y as in Section 2.1.3. Then the probability that τ (x1 , y1 ) = t and y1 = y is given by ft (0, y). Actually a similar rate conservation equation can also be written to compute the distribution of I1 by putting the constraint that Vt (1) = 0.
22
4.2
MMPP arrivals
Let {Yt } be the modulating Markov chain with a finite state space. The notations and assumptions here will be the same as in Section 3.2. The process {(Vt (1), Vt (0), Yt )} is a Markov process. The stability, rates of convergence and functional limit theorems for this process as well as for the waiting time and the queue length process hold as for the single class case in the way explained before. Now we compute the stationary distributions for these processes along with the probability of packet loss for each class. The general approach is the same as in Section 3.2. By writing the rate equations as for the Poisson case (with an extra term), one solves them as in Section 3.2. Thus we obtain the stationary probabilities fi (x, y), gi(x), hi (y). To obtain the probability of packet loss and the waiting time distributions for each class, we need the stationary probabilities at the arrival instants of each class. Then the stationary density of (Vt (1), Vt (0), Yt ) to be (x, y, i) at the arrival instant of a class j packet is, as shown in Section 3.2, (for x > 0, y > 0) λi,j fi (x, y) . P k πk λk,j In this formula, fi (x, y) is replaced by gi (x) or hi (y) if V (0) = 0 or V (1) = 0. Thus, the probability of packet loss of a class j, j = 1, 0, packet is X Zi s 0
λi,j P k λk,j πk
Z 0
s
Z 0
s
fi (x, y)(1 − Bi,j (s − x − y))dxdy+
gi (x)(1 − Bi,j (s − x))dx +
Z 0
s
hi (y)(1 − Bi,j (s − y))dy + Pi (1 − Bi,j (s)) . (16)
Next we obtain the waiting time distribution for class 1. The probability that a packet of class 1 is admitted and experiences a delay of x > 0 is Z s−x X 1 λi (1) fi (x, y)Bi,1(s − x − y)dy + gi (x)Bi,1 (s − x) /(1 − p(1)). P 0 k λk (1)πk i
The waiting time distribution of class 0 is more difficult to obtain. However, it can be obtained by the methods used previously. We know the stationary distribution (x1 , y1, i1 ) of the system state as seen by an admitted class 0 packet. To compute the joint distribution of τ (x1 , y1 , i1 ), the class 0 workload and the state of {Yt } at that time, we can write the rate equations for the transient probabilities for this system, as done in Section 3.2. Then the rest of the procedure remains as before.
23
Acknowledgments This work was performed during the stay of Vinod Sharma as a Visiting Professor at the Helsinki University of Technology.
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13. V. Ramaswami, Matrix analytic methods: A tutorial overview with some extensions and new results, in Matrix Analytic Methods in Stochastic Models, S.R.Chakravarthy and A.S.Alfa (Ed.), Marcel Dekker, N.Y., 1997. 14. J. Roberts, U. Mocci, J. Virtamo (eds.), Broadband Network Teletraffic – Final Report of Action, COST 242, Springer Lect. Notes in Computer Science, 1996. 15. V. Sharma, Reliable estimation via simulation, Queueing Systems, Vol. 19, 1995, 169192. 16. V. Sharma and N.D. Gangadhar, Discrete time queues with time and space priorities, to appear in 3rd IFIP TC6 workshop on Traffic Management and Design of ATM Networks, London, 1999. 17. Y. Takahashi and O. Hashida, Delay analysis of discrete time priority queue with structured inputs, Queueing Systems, Vol. 8, 1991, 149-164. 18. X. Tan and C. Knessl, Integrated representations and asymptotics for infinite and finite capacity queue described by the unfinished work I, II, SIAM J. Applied Math., Vol. 57, 1997, 791-823, 824-870. 19. W. Wagner, Analysis of a multiserver model with nonpreemptive priorities and nonrenewal input, in J. Labetoulle and J.W. Roberts (ed.) Proc. ITC-14, 1995, 757-767.
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