Robert B. Cooper

• Florida Atlantic University

The notation and conventions used in specifying queueing models are described.

We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups , according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups , according to which the server sets up only when work is wa...

We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups, according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups, according to which the server sets up only when work is wait...

Sarkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with exhaustive and gated service disciplines, some ex...

The classical renewal-theory (waiting time, or inspection) paradox states that the length of the renewal interval that covers a randomly-selected time epoch tends to be longer than an ordinary renewal interval. This paradox manifests itself in numerous interesting ways in queueing theory, a prime example being the celebrated Pollaczek-Khintchin...

We consider the classical polling model: queues served in cyclic order with either exhaustive or gated service, each with its own distinct Poisson arrival stream, service-time distribution, and switchover-time (the server’s travel time from that queue to the next) distribution. Traditionally, models with zero switchover times (the server travels at...

Abstract We consider a system of N queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues); and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the...

Certain hash-structured files consist of sequences (chains) of computer memory locations (slots) into which records are inserted, and from which they are later retrieved or deleted. If we assume that the records arrive to the file according to a Poisson process for insertion into a chain (randomly selected by the hash function), and reside in memor...

Using constructive, sample-path arguments, we derive a variety of transform-free results about queue lengths and waiting times for the M/G/1/K queue. In classical analyses of M/G/1/K, it is typical to work with Markov processes obtained by defining the “state” of the system at a time epoch to be the number of customers present and, as supplementary...

Abstract Using a generalization of the classical ballot theorem, Niu and Cooper [7] established a duality relation between the joint distribution of several variables associated with the busy cycle in M/G/1 (with a modified first service) and the corresponding joint distribution of several related variables in its dual GI/M/1. In this note, we gene...

We consider two nonsingular versions of the problem described in the title. For one of these versions, we show by example that neither Jacobi nor Gauss-Seidel iteration is guaranteed to converge; for the other version, we outline a proof that both methods are guaranteed to converge.

This chapter discusses the queueing theory. Queueing theory concerns the construction and analysis of mathematical models of systems that provide service to customers whose arrival times and service requirements are random. The basic queueing model, from which more complicated models can be constructed, consists of three components: (1) the input p...

We consider a loss system with Poisson arrivals and ordered entry, and we suppose that server i has service-time distribution function Fi. Consistent with the results of others, we verify that, in contrast with the classical Erlang loss system (Fi = F for all i), when Fi ≠ Fj then the loss probability is not insensitive to the form of Fi or Fj (eve...

We generalize the classical ballot theorem and use it to obtain direct probabilistic derivations of some well-known and some new results relating to busy and idle periods and waiting times in M/G/1 and GI/M/1 queues. In particular, we uncover a duality relation between the joint distribution of several variables associated with the busy cycle in M/...

Various models and metrics that are based on cognitive theories and assumptions have been proposed in an effort to formalize the concept of programming difficulty. The model and metric proposed here are, instead, based on a problem-space formalization and an information-theoretic measure (entropy) that provides the complexity of a program specifica...

We provide a term-by-term interpretation of Bene\>s's well-known but mysterious inversion of the Pollaczek-Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/1-FIFO with remaining work in M/G/1-LIFO-preemptive resume. In the process, we give a new and simple derivation of some known results for M/G/1-LIFO-preemptiv...

We provide a term-by-term interpretation of Beneš's well-known but mysterious inversion of the Pollaczek–Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/1–FIFO with remaining work in M / G /1–LIFO–preemptive resume. In the process, we give a new and simple derivation of some known results for M/G/1–LIFO–preempti...

This paper considers a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time. As has been noted by other researchers, for several specific models of this type, the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random vari...

We apply a recent decomposition result of Fuhrmann and Cooper for the M/G/1 queue with server vacations to obtain mean waiting times for the following two cyclic queueing models: The server scans at a constant velocity (1) serving work as it is encountered, or (2) collecting work that it serves at the end of each cycle. Model 1 describes token-ring...

It is common for file structures to be divided into equal-length partitions, called buckets, into which records arrive for insertion and from which records are physically deleted. We give a simple algorithm which permits calculation of the average time until overflow for a bucket of capacity n records, assuming that record insertions and deletions...

A queue is a waiting line (like customers waiting at a supermarket checkout counter); queueing theory is the mathematical theory of waiting lines. More generally, queueing theory is concerned with the mathematical modeling and analysis of systems that provide service to random demands. A queueing model is an abstract description of such a system. T...

Takács has shown that, in the M/G/1 queue, the probability P(k | i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k | i) = Qk-i/Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q0 ≠ 0. We augment Takác's theorem by showing that P(k | i) = bk-i...

Takács has shown that, in the M/G/1 queue, the probability P(k | i) that the maximum number of customers present simultaneously during a busy period that begins with i customers present is P(k | i) = Qk –i /Qk, where the Q's are easily calculated by recurrence in terms of an arbitrary Q0 ≠ 0. We augment Takács's theorem by showing that P(k | i) = b...

A well-known theorem of graph theory gives a simple formula for the calculation of the number of spanning trees of a complete graph with n labeled vertices. A well-known proof of this theorem uses a combinatorial identity, related to Abel's generalization of the binomial theorem, that is difficult to prove from first principles. It is the purpose o...

We consider two models, the GI/M/s queue and the M/G/1 queue, in which waiting customers are served in random order. For each model we derive expressions for the calculation of the stationary waiting-time distribution function. Our methods differ from those of previous authors in that we do not use transforms, and consequently our results may be be...

This paper extends the results of a previous paper in which two models of a system of queues served in cyclic order were studied. One model is an exhaustive service model, in which the server waits on all customers in a queue before proceeding to the next queue in cyclic order. The other is a gating model, in which a gate closes behind the waiting...

We study two models of a system of queues served in cyclic order by a single server. In each model, the ith queue is characterized by general service time distribution junction Hi(·) and Poisson input with parameter λi. In the exhaustive service model, the server continues to serve a particular queue until the server becomes idle and there are no u...