General Queueing Model Research Articles

Approximation of The Queue Length Distribution of General Queues

In this paper we develop an approximation formalism on the queue length distribution for general queueing models. Our formalism is based on two steps of approximation; the first step is to find a lower bound on the exact formula, and subsequently the Chernoff upper bound technique is applied to this lower bound. We demonstrate that for the M/M/1 model our formula is equivalent to the exact solution. For the DIM/1 queue, we find an extremely tight lower bound below the exact formula. On the other hand, our approach shows a tight upper bound on the exact distribution for both the ND/D/1 and M/D/1 queues. We also consider the M+Σ NjD/D/1 queue and compare our formula with other formalisms for the Σ NjD/D/1 and M+D/D/1 queues.

Deterministic analysis of queueing systems with heterogeneous servers

Using deterministic (sample-path) analysis, we generalize and extend fundamental properties of systems with “stationary deterministic flows” as introduced by Gelenbe (1983) and Gelenbe and Finkel (1987). Primarily, we provide conditions for stability and instability for general queueing models, and focus attention on multichannel queueing systems with servers that work at different rates. Stability analysis is important in computer applications and usually precedes any further investigation of the system in question. Our results complement and extend those of Gelenbe and Finkel by making weaker assumptions, allowing multichannel facilities with heterogeneous servers, and including more general queueing disciplines such as processor sharing and LCFS-PR. The key to our stability analysis is a deterministic version of the renewal-reward theorem which we call Y = λ X, and a relationship that shows the “operational analysis” definition of average service times, when considered as the observation period t → ∞, coincides with the standard definition of average service times for all stable queueing systems. Our analysis is completely deterministic and avoids any stochastic assumptions about the system under investigation; thus, it provides the practitioner with a method that often leads to a better and deeper understanding of the system under consideration. It also gives a powerful tool to determine which properties of the system are independent of the usually needed probabilistic assumptions. As an illustration, a sample-path relationship that gives the long-run average busy period (cycle) for a general queueing model is given and utilized to derive several well-known results under weaker conditions.

On some queueing models of time division multiple access communication system

In this paper we consider a general queueing model for a time division multiple access communication channel and establish the relationship between the steady-state probability distribution of the number of packets in the buffer at various observation epochs. These results are derived using the theory of embedded Markov chain and discrete state level crossing analysis. We also provide a table describing the relationship between the results given here and the results available in the literature.

The impact of the composition of the customer base in general queueing models

We consider general queueing models dealing with multiple classes of customers and address the question under what conditions and in what (stochastic) sense the marginal increase in various performance measures, resulting from the addition of a new class of customers to an existing system, is larger than if the same class were added to a system dealing with only a subset of its current customer base.Our results enhance our understanding of the dependence of various performance measures with respect to the composition of the customer base. In addition they translate readily into convexity results in an (appropriately defined) arrival rate.

The impact of the composition of the customer base in general queueing models

We consider general queueing models dealing with multiple classes of customers and address the question under what conditions and in what (stochastic) sense the marginal increase in various performance measures, resulting from the addition of a new class of customers to an existing system, is larger than if the same class were added to a system dealing with only a subset of its current customer base. Our results enhance our understanding of the dependence of various performance measures with respect to the composition of the customer base. In addition they translate readily into convexity results in an (appropriately defined) arrival rate.

Letter to the Editor—Comments on “Car Following Headways on Freeways Interpreted by the Semi-Poisson Headway Distribution Model”

We are disturbed by some of the misleading remarks made about our work on the generalized queueing model (GQM) headway distribution by Paul Wasielewski in his recent Transportation Science paper “Car Following Headways on Freeways Interpreted by the Semi-Poisson Headway Distribution Model” (s-PM). It is clear to us from his references to our papers (Cowan, R. J. 1975. Useful headway models. Trans. Res. 9 371–375 and Branston, D. 1976. Models of single lane time headway distributions. Trans. Sci. 10 125–148.) that he had misunderstood the arguments employed in them. Because of this we feel that he misrepresents both our work and the physical basis of the GQM. We acknowledge that Wasielewski is entitled to promote the s-PM, but if he wishes to refer to our work, then he should do so without error. We are outlining some of the errors made by Wasielewski below.

On the eelationship of the transient behaviour of a general queueing model to its idle and busy period distributions

A general queueing model is studied. The main limitation is that bulk arrivals and bulk services are not allowed. Simple relations between the transient state probabilities and the busy and idle period distributions are obtained. The LAPLACE Transform of the idle period distribution always follows from the transient solution. The same is true for one busy period distribution, when intervals between arrivals have negative exponential distributions. On the other hand the transient state probabilities can always be obtained directly from an investigation of a “busy cycle process”.

A General Model for the Performance of Disk Systems

The performance of a disk system is often measured in terms of the length of the waiting line or queue of requests for each of the system's spindles. Thus it is natural to formulate and analyze queueing models of disk systems. While most disk systems have certain characteristics, such as channel interference and concurrent seeks, in common, previous analyses have always been begun from scratch, without exploiting this commonality. We introduce a general queueing model for disk systems, which incorporates the characteristics common to most disk systems, and use it in the approximate analyses of models of the IBM 2314 and 3330 disk systems. Comparisons with simulation statistics show that the approximations made are very good over a wide range of arrival rates and system parameters. We also show how to use the analytic results to investigate performance differences between devices.