Multi Stage Queuing Model in Level Dependent Quasi Birth Death Process

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1 International Journal of Statistics and Systems ISSN Volume 12, Number 2 (217, pp Research India Publications Multi Stage Queuing Model in Level Dependent Quasi Birth Death Process R.Sakthi 1 and V.Vidhya 2 1 Assistant Professor, Department of Science and Humanities, Saveetha School of Engineering, Saveetha University, Chennai, India. 1 Research Scholar, Bharathiar University, Coimbatore, India. 2 Associate Professor, Department of Science and Humanities, VIT, Chennai, Tamil Nadu, India. Abstract This article describes the theoretical approach of the spectral behaviour of a queuing model with generalized structure. Our model is a level dependent quasi-birth and death process for N-stage queuing system using matrix structure in which the necessary and sufficient condition for stationary process is discussed. Keywords: Continuous Time Markov Chain, Level Dependent Quasi Birth Death Process, Hitting Time 1. INTRODUCTION: In this paper, we develop the theoretical approach of the spectral behaviour of LDQBD (Level dependent Quasi Birth Death Process. The application of LDQBD have been acknowledged in WDCMA system model in which performance parameter such as blocking probability of the real time traffic and the expected delay of best effort traffic are computed [3]. The aim of this work is to describe the LDQBD, which have quite complicate structure in the domain of stochastic modelling. The analytic approach of queuing models based on load dependent service rates reviewing back in history to Jackson [4] in which the joint probability distribution of queue lengths with dependent service rate at each station is obtained. Many researchers [1, 9] proposed algorithms to compute the rate matrices of LDQBD based on the matrix continued fraction. Apart from computation the authors [9] compare and state the advantage of their algorithm with the modest algorithm developed by Bright and Taylor [2].

2 294 R.Sakthi and V.Vidhya The discrete and continuous versions of level dependent QBD process described by [5], discusses the necessary and sufficient condition for positive recurrence along with the limiting recurrence along with the limiting distribution for each case. Kroese et al. [6] obtained the sufficient conditions on geometric tail asymptotic for QBD process of a tandem queue with two node, for which the invariant measure of the R matrix is also the invariant measure of the censored Markov chain to level zero. 2. MATHEMATICAL MODEL: This paper is considered with two dimensional continuous time Markov Chain X(t = {(N(t, I(t ; t } where I(t is called the phase of the system at time t, it takes values from the set {,1,2,, m} and N(t is often called the level of the system at time t takes a set of values {,1,2,, N} with N finite. N Number of nodes or stages λ i the arrival rate of jobs from outside to the i th node λ i the overall arrival rate of the jobs at the i th node service rate of the jobs at i th node P ij routing probability, the probability that a job is transferred to the j th node after service completion at the i th node (i j The arrival rate λ i for node i = 1,2,, N of an open network is calculated by adding the arrival rate from outside and the arrival rates from all the other nodes. In the theory of statistical equilibrium the rate of departure is equal to the rate of arrival and over all arrival rate at node i can be written as N λ i = λ i + λ j P ij, for i = 1,2,, N j=1 The level dependent Quasi Birth Death Process X(t on the state space {,1,2,, N} {,1,2,, m} whose generator Q with block tri-diagonally structures generator matrix is given by Q = Q 1 Q Q 12 Q 11 Q 1 Q 23 Q 22 Q 21 Q i,i+1 Q i,i Q i,i 1 ( Q N 2,N 1 Q N 2,N 2 Q N 2,N 3 Q N 1,N Q N 1,N 1

3 Multi Stage Queuing Model in Level Dependent Quasi Birth Death Process 295 where Q, Q 1, Q i,i 1, Q i,i, Q i,i+1 for i = 1,2,, N are (m + 1 (m + 1 matrices. λ 1 λ 1 (λ Q = ( 1 + μ 1 λ 1 μ 1 μ 1 Q 1 = μ 1 ( μ 1 μ 2 μ 2 Q 12 = μ 2 μ 2 ( (λ 2 + μ 2 λ 1 Q 11 = ( (λ 2 + μ 1 + μ 2 λ 2 μ 1 Q 1 = μ 1 ( μ 1 For 2 i N 1 (μ 1 + μ Q i,i+1 = +1 ( +1 (λ i λ 1 Q i,i = ( (λ i λ i+1 Q i,i 1 = ( ( + +1

4 296 R.Sakthi and V.Vidhya and Q N 1,N = Q N 1,N 1 = ( μ N μ N μ N ( μ N μ N (μ N + μ N 1 μ N 1 μ N Theorem: 2.1 The level dependent QBD process with infinitesimal generator matrix Q is positive recurrent iff there exists a strictly positive solution to the system of equations subject to the normalization condition For the nth order row vector π i is given by π (Q 1 + R Q = (1 π ( i= R R 1 R i 1 e = 1 (2 π i = π (R R 1 R i 1, i (3 For i = the equations (2 and (3 which is an empty product leads to the identity matrix I. The elements of these matrices can be represented as follows. The (j, kth element of R i, denoted by [R i ] j,k is the expected sojourn time in state (i + 1, k per unit sojourn in state (i, j, given that the process started in state (i, j. The sequence {R i } is the minimal non-negative solution of the set of equations Q i,i 1 + R i Q i,i + R i (R i+1 Q i,i+1 =, i 1. Bright and Taylor [2] suggested that truncating the infinite series of equations at some level k and then renormalizing to compute an an approximate sub-vector of the form i 1 π i (k = π (k R i, i 1 n= such that π (k satisfies (1 and the normalising condition k π (k (R R 1 R i 1 e = 1. i=

5 Multi Stage Queuing Model in Level Dependent Quasi Birth Death Process 297 The sub-vectors, {π i (k: i }, represents an invariant measure for the limiting distribution of all the states at or below level k. Therefore, for any k, π i (k is an upperbound for π i, and π i (k π i as k. Theorem: 2.2 The QBD process is ergodic, that is positive and components which sum to unity, iff there exists a probability measure ξ such that and ξ (Q 1 + RQ where v = (I + R + R In this case π = ξ ξ v. For m <, the inequality (4 is satisfied iff ξ v < (4 Sp(R < 1 where Sp(R denotes the spectral radius of R. If m is finite and a vector x1 = 1 such that and x(q i,i+1 + Q i,i + Q i,i 1 = for i = 1,2,, N 1, then the QBD is positive recurrent. xq i,i+1 1 < xq i,i 1 1 (5 It follows from the definition that ν R ν (i, j is convergent for all i and j, then z < 1. We shall assume that ω is a 1 invariant measure of R such that ω Rν z ν is finite, which implies z < 1. The following results are immediate consequence of the above discussion. Theorem: 2.3 For a finite irreducible QBD process, if there exists a nonnegative vector ω l 1 and a non-negative number z < 1 such that and ω(q 1 + RQ 12 = ωr = zω then the QBD is ergodic. Also for all invariant i =,1,2, π ni z n = ω i for all n.

6 298 R.Sakthi and V.Vidhya Corollary: 2.4 For a finite irreducible QBD process satisfying(5, the eigenvalue of R are the zeros of the polynomial that lie strictly within the unit circle. Q i,i+1 + zq i,i + z 2 Q i,i 1 For each z, in z < 1, the infinite dimensional tridiagonal matrix Q(z is defined as Q(z = z2 Q i,i 1 + zq i,i + Q i,i+1 z λ i z+1 λ i+1 = ( λ z i z+1 λ i+1 In case of finite, the (N N matrix is defined as Q (N (z = ( λ i z+1 λ i+1 z λ i z+1 z λ i z+1 When m is finite the (m + 1 dimension row vector ω is a left eigen vector of R m corresponding to the eigen vector z with z < 1, iff it satisfies ωq (m+1 (z =. When m is infinite the infinite dimension row vector ω satisfies ωr = zω for z with z < 1, if k=1 ω k q k < and ω satisfies ωq(z =. For finite case of m: As the network queue is assumed to be stable the nonzero eigenvalues of R m given in terms of z lie within the unit circle with Q (m+1 (z =. Thus the eigen value and the corresponding eigen vector of R m is (z, w iff zero is an eigen value of Q (m+1 (z with w as the corresponding eigen vector. Let P i, (x; z, i =,1,2,, N 1 be defined such that P i, (v; z = 1 and P i,n (v; z = P i,n (x; z λ i+1z P i,n 1 (v; z, n 1 satisfying the equations

7 Multi Stage Queuing Model in Level Dependent Quasi Birth Death Process 299 P i, (v; z = 1 z P i,1 (v; z = v + +1 (1 z z P i,2 (v; z = (v + λ i (1 zp i,1 (v; z λ i+1 (1 z z P i,n (v; z = (v + λ i (1 zp i,n 1 (v; z λ i+1 P i,n 2 (v; z, n 3 The following Lemmas are generalization of the Lemma 5.1 and 5.2 of [6] Lemma: 2.5 For each z >, P i,n (v; z has n distinct real zeros v i,n,1 < < v i,n,n Also, v i,n,n > v i,n,n and v i,n,j < v i,n,j < v i,n,j+1, for i =,1,, N 1 and j = 1,2,.., n 1. Lemma: 2.6 The eigen values of Q (n (z are the zeros of P i,n (x; z. The eigenvalues of Q (n (z are the zeros of P i,n (x; z and for each such eigen values x, the corresponding left eigenvector is given by Proof: ( P i, (x; z, P i,1 (x; z,, P i,n 1 (x; z The characteristic polynomial of Q i (1 is XI Q i (1 = X + λ i (1 z. As the Q i (n are tridiagonal, we have XI Q i (2 = (X + λ i (1 z XI Q i (1 z λ i+1 For n 3, XI Q i (n = (X + λ i (1 z XI Q i (n 1 z λ i+1 XI Q i (n 2 Hence, we get ( z n P i,n (x; z is the characteristic polynomial of Q i (n and the eigenvalue of Q (n are the zeros of P i,n (x; z, which proves the first part. The characteristic polynomial Q i (n satisfies XI Q i (n = (X + μi + +1 (1 z XI Q i (n 1 z λ i+1 XI Q i (n 2

8 3 R.Sakthi and V.Vidhya = XI Q i (n λi+1 XI Q i (n 1 = ( z n ( P i,n (x; z λ i+1z P i,n 1 (x; z Hence the eigen values of Q i (n are the zeros of P i,n (x; z. For each eigenvalue x of Q (n, for which P i,n (x ; z = λ i+1z P i,n (x ;z, we have ( P i, (x ; z, P i,1 (x ; z,, P i,n 1 (x ; z (x I Q (n = 3. HITTING DISTRIBUTION In this section the hitting distribution of the set is discussed in brief. F a = {(n 1, m 1 n 1 a} Assuming the QBD process as ergodic the decay rate of the stationary distribution is given by the spectral radius of R. In [7], it was given as 1 lim Sp(R n π k = κ k n n where κ = Ο((1 Sp(R 1 is a constant. It means that the marginal stationary probability that the QBD is in level n decays geometrically with rate Sp(R. For an initial state b S = {,1,2,, N} {,1,2,, m}, define T b be the first return time to b by X and T a the first time N reaches a level a. As a, the limit of T a = inf{n > N[n] a}, T a = inf{n > X[n] = b}. P b {N[T a ] a = s, X[T a ] = x T a < T b } for fixed s and fixe phase I(t. In particular the hitting probability P b follow the same geometric rate of decay as the stationary probabilities π k in (6. (6 4. CONCLUSION AND FUTURE WORK: In this paper we develop the spectral approach of LDQBD process for single server with N(finite stage queue and investigate some theoretical results of the rate matrix. The future work aims to compute the hitting time probability and first passage time distribution of the corresponding QBD process.

9 Multi Stage Queuing Model in Level Dependent Quasi Birth Death Process 31 REFERENCES [1] Baumann, H. and Sandmann, W., 21, Numerical solution of Level Dependent Quasi -Birth and Death Process, Proceedings of the International Conference on Computational Science, ICCS, Procedia Computer Science, 1: [2] Bright, L., and Taylor, P.G., 1995, calculating the equilibrium distribution in level dependent quasi-birth-and death process, Stochastic Models 11(3: [3] Hedge, N., 26, Altman, E., Capacity of multiservice WCDMA networks with variable GoS, Wireless Networks, 12(2: [4] Jackson, J.R., 1963, Jobshop like queuing system, Management Science, 1(1: [5] Kharoufeh, J. P., 211, Level-dependent quasi-birth-and-death processes. In J. Cochran, A. Cox, P. Keskinocak, J.P. Kharoufeh, and J.C. Smith, editors, Wiley Encyclopedia of Operations Research and Management Science, Hoboken, NJ, John Wiley & Sons, Inc. [6] Kroese, D. P., Scheinhardt, W.R.W. and Taylor, P.G., 24, Spectral properties of the tandem Jackson network seen as a quasi-birth and death process, Annals of Applied Probability, 14: [7] Latouche, G. and V. Ramaswami., 1999, Introduction to Matrix Analytic Methods in Stochastic Modeling, vol. 5. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia, PA. [8] Neuts, M.F., 1981, Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach. Dover Publications, Inc., New York. [9] Phung-Duc, T., Masuyama, H., Kasahara, S., and Takahashi, Y., 213, A Matrix Continued Fraction Approach to Multi server Retrial Queues, Annals of Operations Research, Vol. 22(1:

10 32 R.Sakthi and V.Vidhya

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