Sojourn time analysis of a queueing system with two‐phase service and server vacations

We consider a two‐phase service queueing system with batch Poisson arrivals and server vacations denoted by M^X/G₁‐G₂/1. The first phase service is an exhaustive or a gated bulk service, and the second phase is given individually to the members of a batch. By a reduction to an M^X/G/1 vacation system and applying the level‐crossing method to a workload process with two types of vacations, we obtain the Laplace–Stieltjes transform of the sojourn time distribution in the M^X/G₁‐G₂/1 with single or multiple vacations. The decomposition expression is derived for the Laplace–Stieltjes transform of the sojourn time distribution, and the first two moments of the sojourn time are provided. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first‐come‐first‐served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non‐stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999     

Computational analysis of MMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline

This paper studies a queueing system with a Markov arrival process with marked arrivals and PH‐distribution service times for each type of customer. Customers (regardless of their types) are served on a mixed first‐come‐first‐served (FCFS) and last‐come‐first‐served (LCFS) nonpreemptive basis. That is, when the queue length is N (a positive integer) or less, customers are served on an FCFS basis; otherwise, customers are served on an LCFS basis. The focus is on the stationary distribution of queue strings, busy periods, and waiting times of individual types of customers. A computational approach is developed for computing the stationary distribution of queue strings, the mean of busy period, and the means and variances of waiting times. The relationship between these performance measures and the threshold number N is analyzed in depth numerically. It is found that the variance of the virtual (actual) waiting time of an arbitrary customer can be reduced by increasing N. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 399–421, 2000     

Analysis of the finite‐source multiclass priority queue with an unreliable server and setup time

In this article, we study a queueing system serving multiple classes of customers. Each class has a finite‐calling population. The customers are served according to the preemptive‐resume priority policy. We assume general distributions for the service times. For each priority class, we derive the steady‐state system size distributions at departure/arrival and arbitrary time epochs. We introduce the residual augmented process completion times conditioned on the number of customers in the system to obtain the system time distribution. We then extend the model by assuming that the server is subject to operation‐independent failures upon which a repair process with random duration starts immediately. We also demonstrate how setup times, which may be required before resuming interrupted service or picking up a new customer, can be incorporated in the model. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Mean sojourn times in multiclass feedback queues with gated disciplines

In this paper, we investigate multiclass feedback queues with gated disciplines. The server selects the stations by a nonpreemptive priority scheduling algorithm and serves customers in the selected station by either a gated FCFS discipline or a gated priority discipline. We take a rather systematic approach to their mean sojourn times. We define conditional expected sojourn times for all customers and derive their expressions from an analysis of busy periods. Since they are shown to be linear in some components of the system states, their steady state mean values can be derived from simple limiting procedures. These mean values can be obtained from a set of linear equations. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 719–741, 2003.     

On competitive analysis for polling systems

Polling systems have been widely studied, however most of these studies focus on polling systems with renewal processes for arrivals and random variables for service times. There is a need driven by practical applications to study polling systems with arbitrary arrivals (not restricted to time-varying or in batches) and revealed service time upon a job's arrival. To address that need, our work considers a polling system with generic setting and for the first time provides the worst-case analysis for online scheduling policies in this system. We provide conditions for the existence of constant competitive ratios, and competitive lower bounds for general scheduling policies in polling systems. Our work also bridges the queueing and scheduling communities by proving the competitive ratios for several well-studied policies in the queueing literature, such as cyclic policies with exhaustive, gated or l-limited service disciplines for polling systems.     

Optimal admission pricing policies for M/Ek/1 queues

This paper extends the Low-Lippman M/M/1 model to the case of Gamma service times. Specifically, we have a queue in which arrivals are Poisson, service time is Gamma-distributed, and the arrival rate to the system is subject to setting an admission fee p. The arrival rate λ(p) is non-increasing in p. We prove that the optimal admission fee p^* is a non-decreasing function of the customer work load on the server. The proof is for an infinite capacity queue and holds for the infinite horizon continuous time Markov decision process. In the special case of exponential service time, we extend the Low-Lippman model to include a state-dependent service rate and service cost structure (for finite or infinite time horizon and queue capacity). Relatively recent dynamic programming techniques are employed throughout the paper. Due to the large class of functions represented by the Gamma family, the extension is of interest and utility.     

Workload distribution of discrete‐time parallel queues with two servers

We study discrete‐time, parallel queues with two identical servers. Customers arrive randomly at the system and join the queue with the shortest workload that is defined as the total service time required for the server to complete all the customers in the queue. The arrivals are assumed to follow a geometric distribution and the service times are assumed to have a general distribution. It is a no‐jockeying queue. The two‐dimensional state space is truncated into a banded array. The resulting modified queue is studied using the method of probability generating function (pgf) The workload distribution in steady state is obtained in form of pgf. A special case where the service time is a deterministic constant is further investigated. Numerical examples are illustrated. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 440–454, 2000     

Optimal service‐capacity allocation in a loss system

We consider a loss system with a fixed budget for servers. The system owner's problem is choosing the price, and selecting the number and quality of the servers, in order to maximize profits, subject to a budget constraint. We solve the problem with identical and different service rates as well as with preemptive and nonpreemptive policies. In addition, when the policy is preemptive, we prove the following conservation law: the distribution of the total service time for a customer entering the slowest server is hyperexponential with expectation equal to the average service rate independent of the allocation of the capacity. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 81–97, 2015     

Approximations for heavily loaded G/GI/n + GI queues

Motivated by applications to service systems, we develop simple engineering approximation formulas for the steady‐state performance of heavily loaded G/GI/n+GI multiserver queues, which can have non‐Poisson and nonrenewal arrivals and non‐exponential service‐time and patience‐time distributions. The formulas are based on recently established Gaussian many‐server heavy‐traffic limits in the efficiency‐driven (ED) regime, where the traffic intensity is fixed at ρ > 1, but the approximations also apply to systems in the quality‐and‐ED regime, where ρ > 1 but ρ is close to 1. Good performance across a wide range of parameters is obtained by making heuristic refinements, the main one being truncation of the queue length and waiting time approximations to nonnegative values. Simulation experiments show that the proposed approximations are effective for large‐scale queuing systems for a significant range of the traffic intensity ρ and the abandonment rate θ, roughly for ρ > 1.02 and θ > 2.0. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 187–217, 2016     

Minimizing total completion time in two-processor task systems with prespecified processor allocations

We consider the problem of scheduling multiprocessor tasks with prespecified processor allocations to minimize the total completion time. The complexity of both preemptive and nonpreemptive cases of the two-processor problem are studied. We show that the preemptive case is solvable in O(n log n) time. In the nonpreemptive case, we prove that the problem is NP-hard in the strong sense, which answers an open question mentioned in Hoogeveen, van de Velde, and Veltman (1994). An efficient heuristic is also developed for this case. The relative error of this heuristic is at most 100%. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 231–242, 1998     

Markov chain analyses of multiprogrammed computer systems

Most operating systems for large computing facilities involve service disciplines which base, to some extent, the sequencing of object program executions on the amount of running time they require. It is the object of this paper to study mathematical models of such service disciplines applicable to both batch and time-shared processing systems. In particular, Markov queueing models are defined and analyzed for round-robin and foreground-background service disciplines. With the round-robin discipline, the service facility processes each program or job for a maximum of q seconds; if the program's service is completed during this quantum, it leaves the system, otherwise it returns to the end of the waiting line to await another quantum of service. With the foreground-background discipline each new arrival joins the end of the foreground queue and awaits a single quantum of service. If it requires more it is subsequently placed at the end of the background queue which is allocated service only when the foreground queue is empty. The analysis focuses on the efficiency of the above systems by assuming a swap or set-up time (overhead cost) associated with the switching of programs on and off the processor. The analysis leads to generating functions for the equilibrium queue length probabilities, the moments of this latter distribution, and measures of mean waiting times. The paper concludes with a discussion of the results along with several examples.     

Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue

We use the matrix‐geometric method to study the MAP/PH/1 general preemptive priority queue with a multiple class of jobs. A procedure for obtaining the block matrices representing the transition matrix P is presented. We show that the special upper triangular structure of the matrix R obtained by Miller [Computation of steady‐state probabilities for M/M/1 priority queues, Oper Res 29(5) (1981), 945–958] can be extended to an upper triangular block structure. Moreover, the subblock matrices of matrix R also have such a structure. With this special structure, we develop a procedure to compute the matrix R. After obtaining the stationary distribution of the system, we study two primary performance indices, namely, the distributions of the number of jobs of each type in the system and their waiting times. Although most of our analysis is carried out for the case of K = 3, the developed approach is general enough to study the other cases (K ≥ 4). © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 662–682, 2003.     

Optimal scheduling of reader—systems

We consider a reader—writer system consisting of a single server and a fixed number of jobs (or customers) belonging to two classes. Class one jobs are called readers and any number of them can be processed simultaneously. Class two jobs are called writers and they have to be processed one at a time. When a writer is being processed no other writer or readers can be processed. A fixed number of readers and writers are ready for processing at time 0. Their processing times are independent random variables. Each reader and writer has a fixed waiting cost rate. We find optimal scheduling rules that minimize the expected total waiting cost (expected total weighted flowtime). We consider both nonpreemptive and preemptive scheduling. The optimal nonpreemptive schedule is derived by a variation of the usual interchange argument, while the optimal schedule in the preemptive case is given by a Gittins index policy. These index policies continue to be optimal for systems in which new writers enter the system in a Poisson fashion. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 483–495, 1998     

A note on sojourn times in M/G/1 queues with instantaneous,bernoulli feedback

Queueing systems which include the possibility for a customer to return to the same server for additional service are called queueing systems with feedback. Such systems occur in computer networks for example. In these systems a chosen customer will wait in the queue, be serviced and then, with probability p, return to wait again, be serviced again and continue this process until, with probability (1 – p) = q, it departs the system never to return. The time of waiting plus service time, the nth time the customer goes through, we will call his nth sojourn time. The (random) sum of these sojourn times we will call the total sojourn time (abbreviated, sojourn time when there is no confusion which sojourn time we are talking about). In this paper we study the total sojourn time in a queueing system with feedback. We give the details for M/G/1 queues in which the decision to feedback or not is a Bernoulli process. While the details of the computations can be more difficult, the structure of the sojourn time process is unchanged for the M/G/1 queue with a more general decision process as will be shown. We assume the reader is familiar with Disney, McNickle and Simon [1].     

Estimation in single server queues

Moment and maximum likelihood estimates (m.l.e.'s) arc investigated for nonparametric and parametric models for a single server queue observed over a random time horizon, namely, up to the nth departure epoch. Also. m.l.e's of the mean interarrival time and mean service time in anM/M/1 queue observed over a fixed time-interval are studied Limit distributions of these estimates are obtained Without imposing steady state assumptions on the queue-size or waiting time processes.     

The single server queue in discrete time-numerical analysis III

This paper deals with the stationary analysis of the finite, single server queue in discrete time. The following stntionary distributions and other quantities of practical interest are investigated: (1) the joint density of the queue length and the residual service time, (2) the queue length distribution and its mean, (3) the distribution of the residual service time and its mean, (4) the distribution and the expected value of the number of customers lost per unit of time due to saturation of the waiting capacity, (5) the distribution and the mean of the waiting time, (6) the asymptotic distribution of the queue length following departures The latter distribution is particularly noteworthy, in view of the substantial difference which exists, in general, between the distributions of the queue lengths at arbitrary points of time and those immediately following departures.     

Analysis of a machine repair problem with an unreliable server and phase type repairs and services

In this paper we study a machine repair problem in which a single unreliable server maintains N identical machines. The breakdown times of the machines are assumed to follow an exponential distribution. The server is subject to failure and the failure times are exponentially distributed. The repair times of the machine and the service times of the repairman are assumed to be of phase type. Using matrix‐analytic methods, we perform steady state analysis of this model. The time spent by a failed machine in service and the total time in the repair facility are shown to be of phase type. Several performance measures are evaluated. An optimization problem to determine the number of machines to be assigned to the server that will maximize the expected total profit per unit time is discussed. An illustrative numerical example is presented. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 462–480, 2003     

On the busy period of the M/G/1 retrial queue

The M/G/1 queue with repeated attempts is considered. A customer who finds the server busy, leaves the service area and joins a pool of unsatisfied customers. Each customer in the pool repeats his demand after a random amount of time until he finds the server free. We focus on the busy period L of the M/G/1$ retrial queue. The structure of the busy period and its analysis in terms of Laplace transforms have been discussed by several authors. However, this solution has serious limitations in practice. For instance, we cannot compute the first moments of L by direct differentiation. This paper complements the existing work and provides a direct method of calculation for the second moment of L. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 115–127, 2000     

Analysis of a finite‐buffer bulk‐service queue with discrete‐Markovian arrival process: D‐MAP/Ga,b/1/N

Discrete‐time queues with D‐MAP arrival process are more useful in modeling and performance analysis of telecommunication networks based on the ATM environment. This paper analyzes a finite‐buffer discrete‐time queue with general bulk‐service rule, wherein the arrival process is D‐MAP and service times are arbitrarily and independently distributed. The distributions of buffer contents at various epochs (departure, random, and prearrival) have been obtained using imbedded Markov chain and supplementary variable methods. Finally, some performance measures such as loss probability and average delay are discussed. Numerical results are also presented in some cases. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 345–363, 2003.