Shun-Chen Niu

Reliability of Consecutive-k-out-of-n:F System

A reliability diagram with n components in sequence is called consecutive-k-out-of-n:F system if the system fails whenever k consecutive components are failed. This paper presents a recursive formula to compute the exact system reliability, and gives sharp upper and lower bounds for it.

On Johnson's Two-Machine Flow Shop with Random Processing Times

A set of n jobs is to be processed by two machines in series that are separated by an infinite waiting room; each job requires a known fixed amount of processing from each machine. In a classic paper, Johnson gave a simple rule for ordering of the set of jobs to minimize the time until the system becomes empty, i.e., the makespan. This paper studies a stochastic generalization of this problem in which job processing times are independent random variables. Our main result is a sufficient condition on the processing time distributions that implies that the makespan becomes stochastically smaller when two adjacent jobs in a given job sequence are interchanged. We also give an extension of the main result to job shops.

Relating polling models with zero and nonzero switchover times

We consider a system ofN 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 next). We consider two versions of this classical polling model: In the first, which we refer to as the zero-switchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzero-switchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [Queueing Systems 11 (1992) 109—120] and Cooper, Niu, and Srinivasan [to appear in Oper. Res.].

A piecewise-diffusion model of new-product demands

The Bass Model (BM) is a widely-used framework in marketing for the study of new-product sales growth. Its usefulness as a demand model has also been recognized in production, inventory, and capacity-planning settings. The BM postulates that the cumulative number of adopters of a new product in a large population approximately follows a deterministic trajectory whose growth rate is governed by two parameters that capture (i) an individual consumer’s intrinsic interest in the product, and (ii) a positive force of influence on other consumers from existing adopters. A finite-population purebirth-process (re)formulation of the BM, called the Stochastic Bass Model (SBM), was proposed recently by the author in a previous paper, and it was shown that if the size of the population in the SBM is taken to infinity, then the SBM and the BM agree (in probability) in the limit. Thus, the SBM “expands” the BM in the sense that for any given population size, it is a well-defined model. In this paper, we exploit this expansion and introduce a further extension of the SBM in which demands of a product in successive time periods are governed by a history-dependent family of SBMs (one for each period) with different population sizes. A sampling theory for this extension, which we call the Piecewise-Diffusion Model (PDM), is also developed. We then apply the theory to a typical product example, demonstrating that the PDM is a remarkably accurate and versatile framework that allows us to better understand the underlying dynamics of new-product demands over time. Joint movement of price and advertising levels, in particular, is shown to have a significant influence on whether or not consumers are “ready” to participate in product purchase. Subject classifications: marketing: new products, buyer behavior, pricing; probability: diffusion; inventory/production: stochastic, nonstationary demand. Area of review: Manufacturing, Service, and Supply Chain Operations. History: Received August 2003; revision received September 2004; accepted July 2005.

A Single Server Queueing Loss Model with Heterogeneous Arrival and Service

A heterogeneous arrival and service single server queueing loss model is analyzed. The arrival process of customers is assumed to be a nonstationary Poisson process with an intensity function whose evolution is governed by a two-state continuous time Markov chain. Different service distributions for different types of customers are allowed. The explicit loss formula for the model considered is obtained. In a special case, it is shown that as the arrival process becomes more regular the loss decreases. For single server loss systems with renewal arrivals, counterexamples are given to show that regularity of arrival and service distributions do not work to good effect in general. Two sufficient conditions for it to be true are given.

BENES'S FORMULA FOR M/G/1-FIFO 'EXPLAINED' BY PREEMPTIVE-RESUME LIFO

We provide a term-by-term interpretation of Beness well-known but mysterious inversion of the Pollaczek-Khintchine formula. The strategy is to recognize the equality of waiting time in M/G/I-FIFO with remaining work in M/G/1-LIFo-preemptive resume. In the process, we give a new and simple

Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run “state” behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both “actual” and “virtual” delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural “explanations” for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Benešs formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.

Duality and other results for M/G/1 and Gl/M/1 queues, via a new ballot theorem

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 / G /1 and the corresponding joint distribution in GI / M /1. In contrast with the classical derivations of queueing theory, our arguments avoid the use of transforms, and thereby provide insight and term-by-term “explanations” for the remarkable forms of some of these results.

ON THE COMPARISON OF WAITING TIMES IN TANDEM QUEUES

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.

Inequalities between Arrival Averages and Time Averages in Stochastic Processes Arising from Queueing Theory

A common scenario in stochastic models, especially in queueing theory, is that an arrival counting process both observes and interacts with another continuous time stochastic process. In the case of Poisson arrivals, Wolff Wolff, R. W. 1982. Poisson arrivals see time averages. Opns. Res.30 223-231. recently proved that the proportion of arrivals finding the process in some state is equal to the proportion of time it spends there under a lack of anticipation assumption. Inspired by Wolffs approach, in this paper we study the related interesting question of when do we have inequalities between these proportions. We establish two-sided inequalities under the following three assumptions: i the interarrival time distributions are of type NBUE or NWUE, ii the process being observed have monotone sample paths between arrival epochs, and iii the state of the process does not depend on future jumps of the arrival process. These assumptions are typically true in all standard queueing models and hence our results have wide implications. Stochastic inequalities between limiting distributions of interest, when they exist, also follow easily from our main result.