Waiting-Time Distribution of a Discrete-time Multiserver Queue with Correlated. Arrivals and Deterministic Service Times: D-MAP/D/k System

Transcription

1 Waiting-Time Distribution of a Discrete-time Multiserver Queue with Correlated Arrivals and Deterministic Service Times: D-MAP/D/k System Mohan L. Chaudhry a, *, Bong K. Yoon b, Kyung C. Chae b a Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 7000, STN Forces Kingston, Ontario K7K 7B4, Canada b Department of Industrial Engineering, KAIST, Taejon-shi, , Korea Abstract We derive the waiting-time distribution of a discrete-time multiserver queue with correlated arrivals and deterministic (or constant) service times. We show that the procedure for obtaining the waiting-time distribution of a multiserver queue is reduced to that of a single server queue. We present a complete solution to the waiting-time distribution of D-MAP/D/k queue together with some computational results. Keywords: Queues; Discrete-time; multiserver; Markovian Arrival Process; deterministic service. Introduction In discrete-time queues, the time axis is divided into fixed-length time intervals, referred to as slots. Arrivals and services of customers can start or end at slot boundaries only so that the service time is an integer multiple of slots. These assumptions are suitable for modeling ATM (Asynchronous Transfer Mode) networks. Since ATM transmits the information by means of fixed size of ATM cells, discrete-time queues with deterministic (or constant) service times are used in the design of ATM switches. In particular, the Prelude ATM switch with shared-memory architecture, which are polled in a periodic way at a rate of one per time unit (slot), is designed to have the same number of incoming links as the constant service (transmission) time of customers * Corresponding author. Tel: ext. 6460, fax: , Chaudhry-ml@rmc.ca

2 (cells). Further, the potential arrival point of customers is designed to be shifted one slot from one incoming link to the next. Thus, in the Prelude ATM switch, only one arrival occurs in one slot. Gravey et al. [0] have modeled the output buffer of this type of switch by single-server queues with Bernoulli arrival process and constant service time. Bruneel and Wuyts [5] have discussed the same kind of switch which has multiple output channels as an application of the discrete-time multisever queues. However, their models can not accommodate the bursty nature of correlated arrivals of today s telecommunication traffics. As for correlated arrivals, Wittevrongel and Bruneel [7] deal with finite-capacity single server queue with first order correlated arrivals and constant service time in modeling buffer behavior of ATM switches. In this paper, we investigate the waiting-time distribution of a multiserver D- MAP/D/k queue with correlated arrivals and constant service times. Discrete-time Markovian arrival process (D-MAP) has been introduced because of the limitation of Bernoulli processes in modeling correlated arrivals. D-MAP includes first-order correlated arrivals in [7] as a special case. It may be remarked here that the concept of correlated arrivals was first introduced by Chaudhry [7]. The method has recently been extended by several authors and is now known as D-MAP. Since the arrival process of an ATM switch is usually a superposed (multiplexed) process of a large number of traffic sources (incoming links), exact description of the superposed arrival process using D-MAP may require burdensome manipulations of matrices with large state space. In the analyses of the real world applications, in order to reduce the complexity of dealing with matrices with large state space, the superposed arrival process may be approximated by a simpler D-MAP that captures important characteristics of the original process as closely as possible depending on the cost of computation (time) and the required accuracy of the analyses. Blondia [2,3], Onvural [5], and Ferng and Chang [8] discuss the approximation scheme of the superposed process to a simpler process when they deal with the output process of the queue applicable to ATM switch. Though multiserver queues are useful, unfortunately they are hard to handle both 2

3 analytically and computationally. Bruneel and Kim [4] treat a multiserver queue with geometric batch arrivals and constant service times. Xiong et al. [8] discuss a multiserver queue with first-order correlated arrivals and constant service times. Though they deal with the batch arrival case, they only consider queues with service times equal to one slot and focus on the analytical aspects. However, we give a complete and easy solution to the waiting-time distribution of D-MAP/D/k for all values of constant service times using the property of the multiserver queue with constant service times presented in [2]. Note that [5], [0], and [7] also deal with queues having constant service times of arbitrary length - an assumption that is more realistic in applications. Bruneel and Wuyts [5] argue the usefulness of constant service times of arbitrary length against that of one single slot. 2. Waiting-Time Distribution of D-MAP/D/k Queue In our D-MAP/D/k queue, the arrival process of customers is assumed to be governed by D-MAP(C,D), where C and D are, respectively, m-dimensional square matrices implying the transition probability of the phase of underlying Markov chain (UMC) without and with arrivals. C+ D is the transition probability matrix which has the stationary probability vector, π, of the UMC such that π( C+ D) = π, π e =, where e m denotes a column vector of order m with all elements equal to one. The fundamental arrival rate λ of this process is given by λ = πde m. The service time (d) is assumed to be constant and the number of servers is k. If we define the traffic intensity, ρ : = λd k, then we assume that ρ < for the stability of the system. For more details on the D-MAP that is being used here, see Blondia [2]. The basic property of multiserver queues with constant service times is that all servers accept customers periodically if we assume FIFO (first in first out) and cyclic assignment of customers during idle periods (Iversen [2]). Though Iversen uses this property only for continuous-time queueing systems with independent arrivals, it is interesting to know that this property also holds for discrete-time queues with correlated m 3

4 arrivals. With queues having constant service times where this property holds, we show that it is easy to derive the waiting-time distribution of D-MAP/D/k queues. Consider an arbitrary customer who arrives in D-MAP(C,D)/D/k queue. Since all servers of D-MAP(C,D)/D/k queue take customers periodically, the server who serves an arbitrary customer is determined at the arrival epoch of a customer. In view of this, the waiting-time of the arbitrary customer is affected only by the number of customers who will be served by the same server as the arbitrary customer. Thus, in investigating the waiting-time distribution of the arbitrary customer in the D-MAP(C,D)/D/k queue, we only consider the single server who serves the arbitrary customer. Since the arrival process for a specific server in D-MAP(C,D)/D/k queue produces D-MAP( C, D ), see Proposition below, the waiting-time distribution of an arbitrary customer in D- MAP(C,D)/D/k queue is equivalent to that of D-MAP( C, D )/D/ queue, where * C and * D are defined below. Proposition. In D-MAP(C,D)/D/k queue, the arrival process of customers for a specific server produces another discrete-time Markovian arrival process which is D- * MAP( C, D ), where C and D * are km-dimensional square matrices defined by 2 L k- k 2 L k- k C D L L 0 0 * 2 0 C O 0 0 * 2 =, = 0 0 L 0 0 C D. () M M M O O M M M M M M k- 0 0 L C D k- 0 0 L 0 0 k 0 0 L 0 C k D 0 L 0 0 Proof. Since all severs take (or serve) customers periodically, a specific server takes every kth arriving customer in D-MAP(C,D)/D/k queue with FIFO. Therefore, for a specific server, arrivals are considered to occur with k arrival stages. If the current stage is i, i k, it makes a transition into i+ when there is an arrival of a customer in D-MAP(C,D)/D/k queue. On the other hand, if a customer arrives in D-MAP(C,D)/D/k queue when the arrival stage is k, the arrival stage makes a transition into stage. It may 4

5 be remarked that for a specific server, the stage transition from k into means the arrival of a customer who will be served in that specific server. Considering the arrival stage as a new phase of UMC, we can constitute the arrival process for the specific server, which is D-MAP( C, D ). The UMC of D-MAP( C, D ) is two dimensional Markov chain on the state space {( i, j) : i k, j m}, where m denotes the number of phases in the original D-MAP(C,D). It may be remarked that it is instructive to prove Proposition in terms of interarrival times. For details, see Appendix. Thus, when the service times are deterministic, we can easily get the waiting-time distribution of a multiserver queue, D-MAP(C,D)/D/k, from that of a single server queue, D-MAP( C, D )/D/. Remark. The waiting-time distribution of continuous-time multiserver queues with correlated arrivals can be discussed similarly though the numerical work is performed differently from the procedure given below for the discrete-time case. 3. Algorithm for Computing Waiting-Time Distribution In order to obtain the waiting-time distribution of D-MAP(C,D)/D/k queue from the result of D-MAP( C, D )/D/ queue, we give a bit modified version of the algorithm of Alfa and Frigui [] and Frigui et al. [9] who, in turn, use the Neuts' [4] matrix geometric method. The reasons for giving this algorithm here are to (i) correctly state the result given incorrectly in Frigui et al., (ii) change their notation to suit our notation, and finally for the easy readability and applicability of the results of this paper by readers and practitioners. 3.. Preliminary step We describe the Markov chain at the end of a slot. Consider a Markov chain {L n, J n, S n } on the state space {( i, j, s) : i 0, j km, s d}, where L n, J n, and S n denote, respectively, the number of customers in the D-MAP( C, D )/D/ system, the phase of 5

6 UMC in D-MAP( C, D ), and the remaining service time at the nth slot. The transition probability matrix is given by B00 B L B0 A A2 0 0 L P = 0 A 0 A A2 0 L, (2) 0 0 A0 A A2 L M M O O O where its elements are defined below. In order to get the deterministic service time, d, from the phase-type service time represented by ( α, S ), we define 0 Id α = [ 0 L 0 ], S = 0, 0 where I d- denotes (d )-dimensional identity matrix and α is d-dimensional vector. Now, we define the matrices in equation (2) as follows. B = C, B = D α, B = C s, A = C s α, A = D s α + C S, and A = D S, * o * o * o 0 2 where denotes the kronecker product and o = [ ] with T indicating transpose. s 0 L 0 T, d-dimensial vector 3.2. Stationary probability vector x Let x=[ x 0, x, x 2, ] be the stationary probability vector of P, where x 0 is kmdimensional vector and x i s, i, are kmd-dimensional vectors. Then x can be obtained by the following steps. Step. Calculation of G Let G be the minimal nonnegative solution to the matrix polynomial equation, 2 G A AG A G. = The matrix G can be obtained by the following iterative equation, 6

7 2 Gi+ = ( I A0) ( A0 + A2G i ), i 0, where I is the identity matrix and G 0 is a matrix with all elements equal to zero. Step 2. Calculation of x 0 and x Define the matrix B[R] by where R = A2( I A A2G ). B B = B0 A + RA [ ], BR The stochastic matrix B[R] has a km(d+)-dimensional stationary probability vector [ x 0, x ]. Normalize the vector [ x 0, x ] by xe x I R e 0 km + ( ) kmd =, where e i denotes an i-dimensional column vector with all elements equal to one. Remark 2. x 0 and x can also be obtained using the property of the mean recurrence time. In any case, we should compute x separately from x 0 because the dimensions of x 0 and x are different. As stated earlier, Frigui et al. [9] incorrectly state the formula for the calculation of x from x 0 (Eq. (2) in [9]). We get both x 0 and x from the results in Neuts. (For more details, see Neuts [3, pp.25] and [4, pp.38]) Step 3. Finally, the stationary probability vectors, x i s, of P are given by i x + = x R, i. i 3.3. Waiting-time distribution Let y i, i 0, denote the stationary probability vector that a customer sees i customers ahead of him in the D-MAP( C, D )/D/ queue at an arrival epoch. Then y i can be obtained from 7

8 and 0 * ( o y = xd 0 + xd s ), λ where * ( o yi = xd i S+ xi+ D s α ), i, λ * λ denotes the fundamental arrival rate of the D-MAP( C, D ). Finally, we can get w(r), the probability that a customer has to wait exactly r slots before getting service, as follows. where the d-dimensional square matrix, and w(0) = ye, 0 km r () i () = yi( ekm Id) Ω () ed,, i= wr r r Ω () i () r () r o Ω r S s α, is given by () =, r, () i o i Ω () i = ( s α ), i, () i o ( i ) () i Ω ( r) = s αω ( r ) + SΩ ( r ), r i+ and i 2. The computations of probabilities, x i, y i, and w(i) are carried out until the difference between one and each sum of the probabilities is less than ε. For example, the sum of w(i) should satisfy max wi ( ) < ε. i= 4. Numerical Example For numerical examples, the waiting-time distributions of multiserver queues are computed from the results given in the previous sections. Table shows the waitingtime distributions of D-MAP/D/3 and Geom/D/3 as well as the difference between them. The input parameters of D-MAP are given by C =, = D. The parameter of Geometric arrivals is determined in order to get the same traffic intensity as that of D-MAP case. The number of servers is 3 and the constant service time is 0 slots in both cases. We observe from Table that the waiting-time distribution 8

9 of queues with D-MAP has a thicker tail than that of geometric arrivals. This phenomenon is due to the burstiness of the arrival process. It may be remarked here that our numerical results on Geom/D/k queues agree with the results given by Chaudhry et al. [6] who use roots in their analysis. Table 2 shows the performance measures such as the probability that customers have to wait, the mean and the variance of the waiting time. We change the service time and the number of servers while we keep the traffic intensity fixed at ρ = The input parameters used in Table 2 are given by C =, = D. We also observe from Table 2 that the performance measures of D-MAP/D/k queue become better as the number of servers increases. This phenomenon as explained in Iversen [2] indicates both the economy of scale and the regularity of the arrival process. The explanations seem to hold in the discrete-time MAP case as well. 5. Concluding Remarks This paper presents a complete analysis of the waiting-time distribution of the multisever queue with arrivals following D-MAP and constant service times. We show that the waiting-time distribution of the D-MAP/D/k queue can be reduced to the waiting-time distribution of a certain D-MAP/D/ queue. In a similar way, we can divide the servers of D-MAP(C,D)/D/r k into k groups of r servers, that is, the waiting time distribution in D-MAP(C,D)/D/r k is also reduced to that of D-MAP( C, D )/D/r. Acknowledgement The first author acknowledges with thanks the partial financial support (from BK 2 project) and other facilities provided by the Department of Industrial Engineering, KAIST, where he held invited distinguished professorship. This research was also supported (in part) by NSERC. The authors would like to thank the referee and the associate editor for their 9

10 constructive comments which led to a considerable improvement of the paper. Appendix. Interarrival times in D-MAP(C,D) and D-MAP( C, D ) Let τ n and J τ, respectively, denote the instant of time and the phase of UMC at n the arrival of the nth customer in D-MAP(C,D). Then, the joint probability of the interarrival time and the phase of the underlying Markov chain (UMC) is x Pr[ Jτ = j, τ ] [ ],, n n τn x Jτ i n ij x + + = = = C D (3) where [ ] ij denotes the ijth element of a matrix. From equation (3), the matrix generating function of the interarrival times and the phase of UMC in D-MAP(C,D) is given by ( ) I Cz D z. (4) Since a specific server takes every kth arriving customer in D-MAP(C,D)/D/k queue, the probability generating function (PGF) of the interarrival times and the phase of UMC for the specific server is k-fold convolution of equation (4). This results in [( z) z] k I C D. Further, since ( I C) D is the transition probability matrix of UMC at an arrival instant (Herrmann []), the stationary probability vector of UMC at the arrival instant of customers in MAP(C,D), φ, is given by can easily confirm that φ is also the stationary probability vector of successively multiplying both sides of φ( I C) D= φ by φ( I C) D= φφe, =. We [( ) ] k I C D by ( I C) D. Thus, φ is the stationary probability vector of UMC at the arrival instants of customers to the specific server. The PGF of the stationary interarrival times of customers to a specific server in D-MAP(C,D)/D/k queue is [( z) z] k m φ I C D e. From equation (4), the matrix generating function of the interarrival times and the phase of UMC in D-MAP( C, D ) is given by ( ) * I C z D z, where * I is an identity matrix with appropriate dimensions. From the arguments on the inverse of the partitioned matrix in Searle [6, pp.260], we have 0

11 k [( I Cz) Dz] 0 L 0 k * [( z) z] ( z) z I C D 0 L 0 I C D =. (5) M M M ( I Cz) Dz 0 L 0 The stationary probability vector of a stochastic matrix, ( ) * I C D, is given by φ φ 0 L 0. (6) * = [ ] Thus, the PGF of the stationary interarrival times of customers in D-MAP( C, D )/D/ queue is * φ ( I C z) D * ze km. Using equations (5) and (6), we confirm that ( z) z km * * φ I C D e results in [( z) z] k m φ I C D e which is the PGF of the stationary interarrival times of customers to a specific server in D-MAP(C,D)/D/k queue. References [] A.S. Alfa, I. Frigui, Discrete NT-policy single server queue with Markovian arrival process and phase type service, E.J.O.R. 88 (996) [2] C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian Journal of Operations Research, Statistics and Computer Science 32 (3/4) (993) [3] C. Blondia, Statistical multiplexing of VBR sources: A matrix-analytic approach, Performance Evaluation 6 (992) [4] H. Bruneel, B.G. Kim, Discrete-time Models for Communications Systems Including ATM, Kluwer, Boston, 993. [5] H. Bruneel, I. Wuyts, Analysis of discrete-time multiserver queueing models with constant service time, Operations Research Letters 5 (994) [6] M.L. Chaudhry, N.K. Kim, K.C. Chae, Equivalence of bulk-service queues and multiserver queues and their explicit distributions in terms of roots, Technical Report # 0-07, Dept. Industrial Engineering, KAIST (200). [7] M.L. Chaudhry, Queueing problems with correlated arrivals and service through parallel channels, C.O.R.S. 6 (967) [8] H.W. Ferng, J.F. Chang, Connection-wise end-to-end performance analysis of queueing networks with MMPP inputs, Performance Evaluation 43 (200)

12 [9] I. Frigui, A.S. Alfa, X. Xu, Algorithms for computing waiting-time distributions under different queue disciplines for the D-BMAP/PH/, Naval Research Logistics, 44 (997) [0] A. Gravey, J.R. Louvion, P. Boyer, On the Geo/D/ and Geo/D//n queues, Performance Evaluation, 7-25 (990). [] C. Herrmann, The complete analysis of the discrete time finite DBMAP/G//N queue, Performance Evaluation 43 (200) [2] V.B. Iversen, Decomposition of an M/D/r k queue with FIFO into k E k /D/r queues with FIFO, Operations Research Letters 2 (983) [3] M.F. Neuts, Matrix Geometric solutions in stochastic models, The Johns Hopkins University Press, 98. [4] M.F. Neuts, Structured stochastic matrices of M/G/ type and their applications, Marcel Dekker, New York and Basel, 989. [5] R.O. Onvural, Asynchronous Transfer Mode Networks: Performance Issues 2 nd ed., Artech House, Boston/London, 995. [6] S.R. Searle, Matrix algebra useful for statistics, Wiley, New York, 982. [7] S. Wittevrongel, H. Bruneel, Discrete-time queues with correlated arrivals and constant service times, Computers & Operations Research 26 (999) [8] Y. Xiong, H. Bruneel, B. Steyaert, Deriving delay characteristics from queue length statistics in discrete-time queues with multiple servers, Performance Evaluation 24 (996)

13 Table The Distributions of Waiting Times (P(W=t)) (D-MAP/D/3 and Geom/D/3, ρ=0.7879, service time=0) Time (t) DMAP Geom DMAP-Geom Mean Variance

14 Table 2 The Performance Measures of Multiserver Queues (D-MAP/D/k, ρ=0.7909, service time=2k ) The number of servers (k) P(W>0) Mean (W) Variance (W)