Ralph L. Disney

Queueing Networks: A Survey of Their Random Processes

In this paper we review three topics in queueing network theory: queue length processes, sojourn times, and flow processes. In the discussion of the queue length processes we present results for the continuous-time process and several embedded processes. Then we compare continuous-time processes with embedded processes. In considerable generality we present results for mean sojourn times and discuss the distributions of sojourn times. In the discussion of flow processes we present results for various queueing systems. Our bibliography of over 300 references, while not exhaustive, does cover the major papers for the topics considered.

Stationary queue-length and waiting-time distributions in single-server feedback queues

For M/GI/1/oo queues with instantaneous Bernoulli feedback timeand customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek-Khinchine formula for M/GI/1/oo queues without feedback. POISSON ARRIVALS; TIMEAND CUSTOMER-STATIONARY CHARACTERISTICS

QUEUES WITH DELAYED FEEDBACK

Queues with delayed feedback have been little studied in queueing theory. Presented here is a rather complete discussion of such problems including queue length processes, busy period processes and several customer flow processes (e.g., departure processes). The case in which the delay mechanism is an M-server queue is studied in detail but it is shown later that many of the results carry over to a more general delay mechanism.

CROSS CORRELATIONS IN DECOMPOSED STREAMS OF EVENTS WITH APPLICATIONS TO QUEUEING NETWORKS

One can think of a queueing network as a collection of three operations performed on one or more stochastic processes. The order in which these operations are performed is defined by the routing properties of the network. The three operations have been called decomposition, stretching, and recomposition. In queueing networks, decomposition corresponds to splitting the departure process from a given node into subprocesses each of which is to be the arrival process for a next node. The routing properties of the network determine how this splitting is to be made. The departure process from a node is simply a stretched version of the arrival process to that node. The amount of stretching depends on queue disciplines and service characteristics. Recomposition is a merging operation that combines streams of customers from several nodes to form the arrival process to a given node. This operation is determined by the routing properties of the network and a rule, called a recomposition switch, that controls how the several streams are to merge. In nearly all studies of Jackson queueing networks, the decomposition is effected by a multinomial process that allocates a customer emerging from node i and going to node j with a probability p(i,j) independent of all else in the network. Furthermore, in most of the Jackson network studies, recomposition is affected by a superposition rule that acts assentially as a first come-first served rule on a point process. That is, the first customer to arrive to the recomposition switch is the first customer to pass through. Clearly, these are not the only manner in which customers split or merge in more general networks (e.g., production networks or road traffic networks). In this paper we concentrate on the decomposition problem though one example of our study applies to recomposition problems. Previous studies have shown that customer flows in Jackson queueing networks are Markov renewal processes (though perhaps not on countable state spaces) that under some conditions are renewal processes or even Poisson processes. Thus, our paper consists of studying ways of decomposing Markov renewal processes. In particular we let the decomposing process be a random process that may depend on the state of the process being decomposed. Each random processes produced by decomposing a given process has been studied in some detail in the literature. Our interest is in the joint properties of these processes. To obtain tractable results we assume that a given Markov renewal process is decomposed into two subprocesses, though, in principle, the general results go through for decomposing into any finite number of subprocesses. The key question we would like to answer is: What is the joint distribution of the two subprocesses?. It is well known that, in the renewal case, these subprocesses are independent if and only if all three processes are Poisson processes. So that, in general, studying the joint properties of the two subprocesses is of some importance. Answers to the key question above are presently beyond our reach so we lower our sights to study the cross-covariance and cross-corrections between the two subprocesses. There are practical reasons for studying these cross-covariance and cross-correlational properties in addition to the fact that they are tractable. In nearly any queueing network, the flows of customers within the network are auto-correlated as well as cross-correlated. Thus, any statistical analysis of these flows must be prepared to deal with such lacks of independence. Further, many statistical studies of these flows will use correlational methods to study dependencies rather than testing independence directly. Thus, dependence results will more practically be approached statistically through correlation rather than analytically in such things as systems’ simulation. In this paper we will study the cross-correlational properties of 2 Markov renewal processes that are created by the decomposition of a Markov renewal process under the operation of a state dependent switch. By particularization many of our results apply to more commonly occurring situations (e.g., decomposing renewal processes). After establishing some general relations, we apply our results to one simple queueing network (the M/M/1 overflow queue) to show some unexpected properties of customer flow processes in these networks. In particular, this example will show that one can recompose two dependent renewal processes neither of which is a Poisson process to obtain a Poisson process. By time reversal this implies that one can decompose a Poisson process into two subprocesses neither of which is a Poisson process. Of course, the two subprocesses are not independent. This class of problems begins to get at the question: Why do results such as product form solutions work?.