Closed Exponential Networks of Queues with Saturation: The Jackson-Type Stationary Distribution and Its Asymptotic Analysis

Two models of a closed queueing network with saturation are proposed. Given that the “reversibility” condition holds in the first model, the stationary distribution is shown to be of product-form for either of them. Queueing networks with the space of admissible states generated by limited capacities of servers is the most important special variant of the general scheme. The asymptotic analysis of the stationary distribution in the case of a large number of customers is given and shows that a special nonlinear programming problem must be solved to obtain parameters of the limiting distribution. In particular, saturation probabilities are asymptotically expressed through the Lagrangian multipliers dual to corresponding linear restrictions of queue-sizes for servers with limited capacities.

[1]  A. Barbour.  Networks of queues and the method of stages , 1976, Advances in Applied Probability.

[2]  Frank Kelly,et al.  Networks of queues with customers of different types , 1975, Journal of Applied Probability.

[3]  Donald L. Iglehart,et al.  Approximations for the repairman problem with two repair facilities, I: No spares , 1973, Advances in Applied Probability.

[4]  N. Prabhu.  Transient Behaviour of a Tandem Queue , 1967 .

[5]  Hosei Uniuersity,et al.  ON A TANDEM QUEUE WITH BLOCKING , 1963 .

[6]  M. Neuts.  Two queues in series with a finite, intermediate waitingroom , 1968, Journal of Applied Probability.

[7]  J. Kingman.  Markov population processes , 1969, Journal of Applied Probability.

[8]  F. Kelly,et al.  Networks of queues , 1976, Advances in Applied Probability.

[9]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[10]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[11]  Alan G. Konheim,et al.  A Queueing Model with Finite Waiting Room and Blocking , 1976, JACM.

[12]  B. Avi-Itzhak,et al.  A Sequence of Two Servers with No Intermediate Queue , 1965 .

[13]  E. Levitin.  Differentiability with respect to a parameter of the optimal value in parametric problems of mathematical programming , 1976, Cybernetics.

[14]  Hisashi Kobayashi,et al.  Application of the Diffusion Approximation to Queueing Networks I: Equilibrium Queue Distributions , 1974, JACM.

[15]  W. J. Gordon,et al.  Closed Queuing Systems with Exponential Servers , 1967, Oper. Res..

[16]  D. R. Fulkerson,et al.  Flows in Networks. , 1964 .