INCORPORATING OPEN QUEUEING MODELS INTO CLOSED QUEUEING NETWORK ALGORITHMS

ANDRE BENJAMIN BONDI, Purdue University

Abstract

Many closed queueing networks contain servers with first come-first served (FCFS) queueing discipline whose service time distributions are not exponential (e.g., a central processing unit) or which do not begin serving a waiting job immediately after the previous one has departed (e.g., a local area network gateway). Although it is often possible to formulate open-queue models of such servers that accurately predict their expected response times and queue lengths, exact models of closed networks containing these servers are usually intractable, and when available, cannot be solved using existing efficient algorithms such as convolution or mean value analysis (MVA) because the networks do not have product form. To overcome this problem, we would like to develop a robust and computationally tractable means of incorporating accurate models of general servers into either of these two solution methods. Incorporation of open models into solution methods for closed models is not straightforward. Queue lengths in open systems, unlike those in closed systems, are unconstrained. A study of closed networks with non-exponential servers reveals that the queue length of a bottleneck server under moderate and heavy loads, unlike the queue length of an M/G/1 server in isolation, is a decreasing function of the service time's coefficient of variation. Increased service time variance at the bottleneck server leads to increased inter-arrival time variance at other servers. The resulting increase in the other servers' queue lengths draws jobs away from the bottleneck. The phenomenon is predicted both by Marie's approximate algorithm for closed queueing networks with general Coxian servers and by exact non-product form solutions of small systems. An implication of the phenomenon is that approximation methods will not be reliable if they are based on the assumption that queue lengths always increase with the service time's coefficient of variation. Marie's algorithm accounts for the effect of high inter-arrival time variability by assuming that arrival processes at each server are state-dependent. It also solves the problem of efficiently modeling the performance of a closed network with a central processing unit having a high coefficient of variation. We use Marie's algorithm as the basis for a proposed framework for iteratively incorporating existing models of general servers into the convolution algorithm for product form networks; the framework reduces to his original algorithm if the server has a Coxian distribution and does not block jobs when it is idle. The framework has been used to incorporate a modified, validated version of Berry and Chandy's approximate model of a token ring with non-exhaustive service into closed queueing network models of the hosts it connects. The integrated system model has predicted performance measures close to those produced by simulations for various sets of model parameters.

Degree

Ph.D.

Subject Area

Computer science

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