Overload Analysis of the PH/PH/1/K Queue and the Queue of M/G/1/K Type with Very Large K

Transcription

1 Overload Analysis of the PH/PH/1/K Queue and the Queue of M/G/1/K Type with Very Large K Attahiru Sule Alfa Department of Mechanical and Industrial Engineering University of Manitoba Winnipeg, Manitoba Canada R3T 5V6 alfa@cc.umanitoba.ca and Yiqiang Q. Zhao Department of Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9 zhao@io.uwinnipeg.ca September 2, 2004 Abstract We consider the PH/PH/1/K queue and the queue of M/G/1 type with very large buffer size K and operating in overload condition. By rotating the resulting transition matrix through 180 o we turn it into a queue with underload condition. By approximating the latter queue by an infinite buffer system we are able to study it using the matrix-geometric method. A procedure for estimating the blocking probabilities is presented. We apply the procedure to a telephone system with both patient and impatient customers. 1 Introduction The interest in the performance analysis of finite buffer queueing models carried out in most telecommunication systems is usually in the overload cases, i.e. when traffic intensities are greater than unity. One of the interesting measures is the blocking or loss probability. There is a large volume of references on this topic, many of which can be found in the Ph.D. thesis 1

2 by Gouweleeuw 2. As such the usual approximating approach for finite buffer queues which is to use the results of infinite buffer queues would not be applicable since the infinite system is not stable. Usually one is left with the only choice which is to carry out the analysis using standard finite Markov chain analysis approach. This approach is extremely time consuming especially when the buffer size is very large; for example buffer sizes in the thousands are not uncommon (Zhao and Alfa 11 and Zhao and Campbell 10). In a recent paper, Zhao 12 presented the method of rotation which can be used to accurately estimate the steady state distributions of such systems when they are the M/G/1 and GI/M/1 queues or their variants. The method is based on the premise that: if a single server queue with a finite buffer is transformed to another queue whose interarrival time is the service time of the original queue and the service time is the interarrival time of the original queue, then the traffic intensity of the new queue is the reciprocal of the traffic intensity of the original queue. The transition matrix of the Markov chain representing the new queue is the transition matrix representing the Markov chain of the old queue but rotated through 180 o. The resulting invariant vector of the new matrix is the mirror image of that of the original matrix. If the buffer size is very large, which is the case that normally poses the major computational problem, we can then rotate the system of such queue with high overload, thereby resulting in an underload case and then approximating that by an infinite buffer model. In this paper, we show how the method can be extended to quasi-birth-and-death models such as the PH/PH/1/K queues and to the queues of M/G/1/K type, which includes an interesting application to a telephone system with both patient and impatient customers. 2 The main result Since the rotation procedure is the same for both the PH/PH/1/K queue and the queue of M/G/1/K type, we only go details for the PH/PH/1/K queue. 2.1 The P H/P H/1/K queue Since we are only interested in results in steady state, we will consider the generator of the model, or the uniformized trasition matrix. Alternatively, we use the discrete time 2

3 representation for a phase type distribution. We consider a finite buffer single server queue in which interarrival times have the phase type distribution represented by (α,t) of dimension n with mean λ 1, and service times which are also of the phase type distribution represented by (β,s) of dimension m with mean µ 1. In the representation of a phase type distribution, the first variable is a vector and the second is a square matrix (Neuts 7). Let the buffer size be K. The PH/PH/1/K type queue occurs frequently in manufacturing systems. As pointed out by Buzacott and Shanthikumar 1 phase-type distributions do naturally occur for service times in manufacturing. For example service to a job involves performing several tasks each of which has an exponential time distribution. As for phase type arrivals it is well known that several types of interarrival times (which are independent and identically distributed) in most practical queueing systems can be represented or approximated by phase type distributions. In addition the buffer spaces in such systems are usually finite. As such PH/PH/1/K queues do occur frequently in manufacturing systems. Now, consider two other phase type distributions represented by (α,t ) of dimension n and (β,s ) of dimension m. We assume that these two are related to the first two as follows: and α i = α n i+1 and T i,j = T n i+1,n j+1, 1 i,j n (1) β i = β m i+1 and S i,j = S m i+1,m j+1, 1 i,j m. (2) It is clear that the two phase types represented by (α,t) and by (α,t ) are stochastically equivalent, similarly those represented by (β,s) and by (β,s ). We define T 0 = e Te, T 0 = e T e, S 0 = e Se, and S 0 = e S e, (3) where e is a column vector of ones. The case we are interested in is such that λ/µ > 1, i.e. the overload case. Note that the buffer size K <. The state space describing the Markov chain representing the P H/P H/1/K queue is given as = {(0,j) (i,j,k), 1 i K + 1, 1 j n, 1 k m}, where (0,j) represents 3

4 the states when the system is empty and the arrival is in phase j and (i,j,k) represents when there are i customers in the system, the arrival is in phase j and service is in phase k. The transition matrix P representing this Markov chain is given as where, P = B 00 B 01 B 10 A 1 A 0 A 2 A 1 A 0 A 2 C, (4) B 00 = T, B 01 = (T 0 α) β, B 10 = T S 0, A 0 = (T 0 α) S, (5) A 1 = (T 0 α) (S 0 β) + T S, A 2 = T (S 0 β), C = A 1 + A 0. (6) Let x = x 0, x 1,, x K+1 be the probability invariant vector associated with P. Then we have xp = x, and xe = 1. (7) Our interest is to compute the blocking probability P b = x K+1 e. Once again, the system is overloaded which implies that (see Neuts 7) πψ < 1, where π = πa, πe = 1, A = A 0 + A 1 + A 2 and ψ = A 1 e + 2A 2 e. If we now rotate P to obtain where, P = C A 2 A 0 A 1 A 2 A 0 A 1 A 2 A 0 A 1 B10 B01 B00, (8) B 00 = T, B 01 = (T 0 α ) β, B 10 = T S 0, A 0 = (T 0 α ) S, (9) A 1 = (T 0 α ) (S 0 β ) + T S, A 2 = T (S 0 β ), C = A 1 + A 0. (10) Let y = y 0, y 1,, y K+1 be the probability invariant vector associated with P. The following lemma provides the relationship between x and y. 4

5 Lemma 1 Let X t be a Markov chain with state space {(i,j),i = 0, 1, 2,,j = 1, 2,,n} and transition matrix P = P 00 P 01 P 0K P 10 P 11 P 1K... P K0 P K1 P KK, (11) where P ij = (P ij (r,s)) are matrices of size n n. Define Y t = (K + 1,n + 1) X t, which means that Y t = (K + 1 i,n + 1 j) if X t = (i,j). Then, the transition matrix P of the Markov chain Y t given by P = P 00 P 01 P 0K P 10 P 11 P 1K... PK0 PK1 PKK, (12) where P ij = (P ij(r,s)) with P ij(r,s) = P K+1 i,k+1 j (n + 1 r,n + 1 s) is the rotation of P by 180 o. The probability invariant vectors x = x 0,x 1,,x K and y = y 0,y 1,,y K corresponding to X t and Y t, respectively, are related by or y i is the rotation of x K+1 i. y i (j) = x K+1 i (n + 1 j), (13) The proof of the lemma is clear if one notices that the rotation P of the transition matrix P is the transition matrix for the Markov chain obtained by renumbering the states in the state space of the original Markov chain. An immediate consequence is Corollary 1 Let x = x 0, x 1,, x K+1 and y = y 0, y 1,, y K+1 be the probability invariant vectors associated with P and P given in 4 and 8, respectively, then y i is the rotation of x K i+1. By noticing the details in the transition matrix P in 8, we find that the Markov chain described by this transition matrix is that of another PH/PH/1/K queue in which the interarrival time has a phase type distribution represented by (β,s ) and the service time is a phase type distribution represented by (α,t ). Let π = π A, π e = 1 and A = A 2 + A 1 + A 0, then π im+v = π (n i)m+m v, i = 0, 1,,n 1;v = 1, 2,,m and A i,j = A n i+1,m j+1. Hence π ψ = πψ, where ψ = A 1e + 2A 2e and π i are the elements of π. 5

6 Theorem 1 π δ > 1, where π = π A, π e = 1, A = A 2+A 1+A 0 and δ = A 1e+2A 0e. Proof δ = A 1e + 2A 0e = A 1e + 2(A e A 1e A 2e) = 2e A 1e 2A 2e = 2e ψ. Multiplying both sides by π we obtain π δ = 2π e π ψ = 2 πψ. Noting that πψ < 1 then we have π δ > 1. We can now approximate this Markov chain by an infinite state Markov chain with the following transition matrix P = C A 2 A 0 A 1 A 2 A 0 A 1 A 2. (14) This Markov chain is stable because according to Theorem 1 its traffic intensity is µ/λ < 1. Let z = z 0, z 1, be the invariant vector associated with P. The convergence of y to z has been established by Li and Zhao 5. Now, we can approximate x K+1 by z 0. We use matrix-geometric approach to compute the relevant parameters (Neuts 7). First we notice that the matrix R is the minimal non-negative solution to the matrix quadratic equation R = A 2 + RA 1 + R 2 A 0. (15) R can be computed using various available algorithms; for example, a logarithmic reduction algorithm by Latouche and Ramaswami 4. Having obtained R we can then calculate z i recursively as follows: z i+1 = z i R, i 1 or z i+1 = z 1 R i, i 0. (16) Therefore, the key computation left now is the boundary probability vector (z 0, z 1 ). Define the matrix BR as C A BR = 2 A 0 A 1 + RA 0 6. (17)

7 According to Neuts 7, the vector (z 0, z 1 ) can be computed by solving (z 0, z 1 ) = (z 0, z 1 ) BR subject to the normalization condition: (z 0 + z 1 (I R) 1 )e = 1. (18) The blocking probability P b of the original overflow queue, which is given by x K+1 e is now approximated by z 0 e. Even though our interest is mainly in the blocking probabilities every other element of the invariant vector can also be reasonably approximated. We have z K+1 i,n j+1,m v+1 y K+1 i,n j+1,m v+1 = x i,j,v, (19) where z = z 0,z 1,, z i = z i,1,1,,z i,1,n,z i,2,1,,z i,j,v,,z i,n,m, y = y 0,y 1,,y K+1, y i = y i,1,1,,y i,1,n,y i,2,1,,y i,j,v,,y i,n,m, (20) x = x 0,x 1,,x K+1, x i = x i,1,1,,x i,1,n,x i,2,1,,x i,j,v,,x i,n,m. We now present some illustrative examples. In example 1 and example 2, we let the buffer space be 100. Example 1 Let the service time be a phase distribution represented by β = , S = We consider the following cases: Case 1: α = 1 0, T = ; Case 2: α = 1 0, T = Case 3: α = 1 0, T = Case 4: α = 1 0, T = Case 5: α = 1 0, T = ; ; 7 ;

8 We report the difference between the blocking probabilities as calculated using the transition matrix P and using the approximation. The true blocking probabilities reported in all tables were computed by solving the stationary equations using the highly numerically stable GTH algorithm (Grassmann, Taksar and Heyman 3). Table 1: Difference in Blocking Probabilities for the Two Methods for Example 1. Case ρ P b difference in blocking prob Example Let the service time be a phase distribution represented by β = , S = We consider the arrival processes in Example 1 and label them Cases The resulting differences in the blocking probabilities are shown in Table 2 below. Table 2: Difference in Blocking Probabilities for the Two Methods for Example 2. Case ρ P b difference in blocking prob The queue of M/G/1/K type In this subsection, we apply the rotation procedure to a telephone switch system, where the impact of the presence of impatient customers on the system under overloaded conditions will be studied. This same system was first studied in Zhao and Alfa 11 under a stable condition. They were not able to perform an overload analysis. For this purpose, we need consider a discrete time Markov chain of M/G/1/K type. Once again, we are interested in the performance measures for the model of finite buffer K with K very large. This type of Markov chains can be treated as a truncated version of the Markov chains of M/G/1 type 8

9 studied by Neuts 8. More specifically, the transition matrix of a Markov chain of M/G/1/K type is given by P = B 0 B 1 B 2 B K 1 i K B i A 0 A 1 A 2 A K 1 i K A i A 0 A 1 A K 2 i K 1 A i A 0 A K 3 i K 2 A i.... A 0 i 1 A i, (21) where all submatrices B i and A i are of size (m + 1) (m + 1). We assume that P and A = i=0 A i are irreducible. Let π be the probability invariant vector for A. Define β = i=1 ia i e. We assume that the system is operated under the overloaded condition: πβ > 1. Let x = x 0, x 1,, x K+1 be the invariant vector associated with P. Then we have xp = x and xe = 1. One of our interests is to compute the blocking probability defined by P b = x K+1 e by using the rotation procedure. For the telephone system, a 0 a 1 a 2 a m 1 a m a m+j a 0 a 1 a 2 a m 1 a m a m+j 0 a 0 a 1 a m 2 a m a m 1+j B 0 = 0 0 a 0 a m 3 a, B m 2 j = a m 2+j a 0 a a 1+j for j = 1, 2,, and A 0 = b 0 b 1 b 2 b m 1 b m , A 1 = and b m+j a m 1+j A j = a m 2+j a j for j = 2, 3,. In the above matrices, for k = 0, 1, 2,, a k = ak k! e a and b k = bk k! e b b m+1 a 0 a 1 a 2 a m 1 a m 0 a 0 a 1 a m 2 a m a 0 a 1,

10 Let E(T) be the average service time of a customer. We are then interested in the overloaded case: λe(t) > 1. The blocking probability P b = y 0 e. In the above equation, y = (y 0,y 1,,y K ) is the invariant probability vector of the rotated Markov chain i 1 A i A 0 i 2 A i A 1 A 0 P =..., (22) i K A i A K 1 A K 2 A 1 A 0 i K Bi BK 1 BK 2 B1 B0 where A i and B i are rotations of A i and B i, respectively. The infinite version P of the rotated transition matrix is given by i 1 A i A 0 i 2 A i A 1 A 0 P = i 3 A i A 2 A 1 A which is of GI/M/1 type. In general, this Markov chain of GI/M/1 type is stable since π β = πβ > 1 where A = i=0 A i and β = i=1 ia ie. According to Li and Zhao 5, P is block-monotone and therefore any block-augmentation converges. As a consequence, P converges to P in the sense that y converges to the invariant vector z associated with P. Since the buffer size K in our case is large, z approximates y. Therefore, the blocking probability can be approximated using z 0 e. z 0 is the invariant vector of BR = k=0 R k B k with B k = i=k+1 A i and R the rate matrix and normalized by z 0 (I R) 1 e = 1. For the telephone system, notice that A 0 = ωβ with ω = (0, 0,,0,b m ) a column vector, where b m = m i=0 b i, and β = (b m,b m 1,,b 0 )/b m. It follows from the results in Ramaswami and Latouche 9, also see Liu and Zhao 6, that 1 R = A 0 I η i 1 A i, i=1 or R = ωξ, where 1 ξ = β I η i 1 A i, (23) i=1 and η = ξω is the maximal eigenvalue of R. Since only the last element in ω is not zero, all rows of R are zero except the last one, which is denoted by r = (r 0,r 1,,r m ) with 10,

11 r m = η. For our case, η can be easily found by computing the positive root, which is less than 1, of determinant det(η k=0 η k A k ) (see Liu and Zhao 6). After finding η, we now can compute ξ using (23). It leads to the following recursive determination. ξ m = η b m, ξ m 1 ξ m k = ξ m b 0 /b m a 0, = (1 a 1)ξ m k+1 a 2 ξ m k+2 a k 1 ξ m 1 b k 1 /b m a 0, k = 2, 3,,m. Finally, r k = b m ξ k, k = 0, 1,,m. In the following, we will use the same parameter values used in Zhao and Alfa 11, that is, the buffer size m = 10 for patient customers, the buffer size K = 100 for impatient customers, the service time a = 1.5 for a patient customer, and the service time b = 1 for an impatient customer. The overload condition can be proved to be λ > 0.68 (see Zhao and Alfa 11). In Table 3 and Table 4, r, and the true blocking probability P b and the error, which is the difference of the blocking probabilities between the true one and the approximation, are provided for different values of λ, respectively. Table 3: The Last Row of the Rate Matrix. λ r 0 r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r

12 Table 4: The True Blocking Probability and the Error of Approximation. λ True P b Error < < < < < Acknowledgements: The authors acknowledge that this work was supported by research grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) and also thank the referees for their valuable comments. References 1 Buzacott, J. A. and Shanthikumar, J. G. (1993), Stochastic Models of Manufacturing Systems, Prentice Hall, Englewood Cliffs, N.J. 2 Gouweleeuw, F.N. (1996), A General Approach to Computing Loss Probabilities in Finite-Buffer Queues, Ph.D. Thesis, Free University, Amsterdam. 3 Grassmann, W.K., Taksar, M.I. and Heyman, D.P. (1985), Regenerative analysis and steady state distributions for Markov chains, Operations Research, Vol. 33, Latouche, G. and Ramaswami, V. (1993), A logarithmic reduction algorithm for quasibirth-death processes, J. Appl. Prob., Vol. 30, Li, H. and Zhao, Y.Q. (1999), Stochastic block-monotone matrix and approximating the stationary distribution of infinite Markov chains, accepted by Stochastic Models. 6 Liu, D. and Zhao, Y.Q. (1996), Determination of explicit solution for a general class of Markov processes, Matrix-Analytic Methods in Stochastic Models, S. Chakravarthy and A. S. Alfa, Eds, Marcel Dekker, Neuts, M. F. (1981), Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach, The Johns Hopkins University Press, Baltimore. 8 Neuts, M. F. (1989), Structured stochastic matrices of M/G/1 type and their applications, Marcel Dekker Inc., New York. 12

13 9 Ramaswami, V. and Latouche, G. (1986), A general class of Markov processes with explicit matrix-geometric solutions, OR Spektrum, , Vol Zhao, Y. Q. and Campbell, L. L. (1995), Performance analysis of a multibeam packet satellite system using random access techniques, Performance Evaluation, Vol. 22, Zhao, Y. Q. and Alfa, A. S. (1995), Performance analysis of a telephone system with patient and impatient customers, Telecommunication Systems, Vol. 4, Zhao, Y. Q. (1998), Overload analysis for Markovian models, System Sci. and System Eng., accepted. 13