Clearing Models for M/G/1 Queues

We consider M/G/1-type queueing systems with disasters, occurring at certain random times and causing an instantaneous removal of the entire residual workload from the system. After such a clearing, the system is assumed to be ready to start working again immediately. We consider clearings at deterministic equidistant times, at random times and at crossings of some prespecified level, and derive the stationary distribution of the workload process for these clearing times and some of their combinations.     

Analysis of an M/G/∞ queue with batch arrivals and batch-dedicated servers

We analyze an M/G/∞ queue with batch arrivals, where jobs belonging to a batch have to be processed by the same server. The number of jobs in the system is characterized as a compound Poisson random variable through a scaling of the original arrival and batch size processes.     

The M/G/1 Queue with Finite Workload Capacity

We consider the M/G/1 queueing system in which customers whose admission to the system would increase the workload beyond a prespecified finite capacity limit are not accepted. Various results on the distribution of the workload are derived; in particular, we give explicit formulas for its stationary distribution for M/M/1 and in the general case, under the preemptive LIFO discipline, for the joint stationary distribution of the number of customers in the system and their residual service times. Furthermore, the Laplace transform of the length of a busy period is determined. Finally, for M/D/1 the busy period distribution is derived in closed form.     

A sample path relation for the sojourn times in G/G/1−PS systems and its applications

For the single server system under processor sharing (PS) a sample path result for the sojourn times in a busy period is proved, which yields a sample path relation between the sojourn times under PS and FCFS discipline. This relation provides a corresponding one between the mean stationary sojourn times in G/G/1 under PS and FCFS. In particular, the mean stationary sojourn time in G/D/1 under PS is given in terms of the mean stationary sojourn time under FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of these results suggest an approximation of the mean stationary sojourn time in G/GI/1 under PS in terms of the mean stationary sojourn time under FCFS. Mathematics Subject Classification (MSC 2000) 60K25· 68M20· 60G17· 60G10 This work was supported by a grant from the Siemens AG.     

Fluid model driven by an M/G/1 queue with multiple exponential vacations

This paper studies a fluid model driven by an M/G/1 queue with multiple exponential vacations. By introducing various vacation strategies to the fluid model, we can provide greater flexibility for the design and control of input rate and output rate. The Laplace transform of the steady-state distribution of the buffer content is expressed through the minimal positive solution to a crucial equation. Then the performance measure-mean buffer content, which is independent of the vacation parameter, is obtained. Finally, with some numerical examples, the parameter effect on the mean buffer content is presented.     

Busy Periods of Poisson Arrival Queues with Loss

We consider two queues with loss, one is the finite dam with Poisson arrivals and the other is the M/G/1 queue with impatient customers. We use the method of Kolmogorov's backward differential equation and construct a type of renewal equation to obtain the Laplace transform of busy(or wet) period in both queues. As a consequence, we provide the explicit forms of expected busy periods.     

Subexponential asymptotics of the stationary distributions of M/G/1-type Markov chains

This paper studies the subexponential asymptotics of the stationary distribution of an M/G/1-type Markov chain. We provide a sufficient condition for the subexponentiality of the stationary distribution. The sufficient condition requires only the subexponential integrated tail of level increments. On the other hand, the previous studies assume the subexponentiality of level increments themselves and/or the aperiodicity of the G-matrix. Therefore, our sufficient condition is weaker than the existing ones. We also mention some errors in the literature.     

Martingale Methods for Analysing Single-Server Queues

In this paper we presents a martingale method for analysing queues of M/G/1 type, which have been generalised so that the system passes through a series of phases on which the service behaviour may differ. The analysis uses the process embedded at departures to create a martingale, which makes possible the calculation of the probability generating function of the stationary occupancy distribution. Specific examples are given, for instance, a model of an unreliable queueing system, and an example of a queue-length-threshold overload-control system.     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time

We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM ^[x]/G/1 queue with batch arrivals.     

Stationary distribution of queue length in G / M / 1 queue with two-stage service policy

We consider a G / M / 1 queue with two-stage service policy. The server starts to serve with rate of μ₁ customers per unit time until the number of customers in the system reaches λ. At this moment, the service rate is changed to that of μ₂ customers per unit time and this rate continues until the system is empty. We obtain the stationary distribution of the number of customers in the system.     

Algorithmic approximations for the busy period distribution of the M/M/c retrial queue

In this paper we deal with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busy period distribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busy period moments. The corresponding distributions for the total number of customers served during a busy period are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/c retrial queue.     

A note on the optimality of the N- and D-policies for the M/G/1 queue

This note considers the N- and D-policies for the M/G/1 queue. We concentrate on the true relationship between the optimal N- and D-policies when the cost function is based on the expected number of customers in the system.     

The Virtual Waiting Time of the M/G/1 Queue with Impatient Customers

The M/G/1 queue with impatient customers is studied. The complete formula of the limiting distribution of the virtual waiting time is derived explicitly. The expected busy period of the queue is also obtained by using a martingale argument.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

Design and G connection of developable surfaces through Bézier geodesics

For potential application in shoemaking and garment manufacture industries, the G¹ connection of (1, k) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bézier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G¹ connection of developable surfaces through abutting cubic Bézier geodesics and give some examples.     

The impact of production time variability on make-to-stock queue performance

In this paper, we exploit the distributional Little’s law to obtain the steady-state distribution of the number of customers in a GI/G/1 make-to-stock queueing system. Non-exponential service times in make-to-stock queue modeling are usually avoided or at best, considered in approximations due to difficulties in developing an exact method. By providing a numerical solution of the GI/G/1 make-to-stock queue, we observed the impact of production time variability on optimal inventory control policies. The numerical results prove the degree of errors in the results if an exponential service time distribution were assumed instead of the actual distribution.     

Analysis of the M X /G/1 Queue Under D-Policy

Abstract

We study the queue length of the M ^X /G/1 queue under D-policy. We derive the queue length PGF at an arbitrary point of time. Then, we derive the mean queue length. As special cases, M/G/1, M ^X /M/1, and M/M/1 queue under D-policy are investigated. Finally, the effects of employing D-policy are discussed.     

Approximations for the delay probability in the M/G/s queue

This paper develops approximations for the delay probability in an M/G/s queue. For M/G/s queues, it has been well known that the delay probability in the M/M/s queue, i.e., the Erlang delay formula, is usually a good approximation for other service-time distributions. By using an excellent approximation for the mean waiting time in the M/G/s queue, we provide more accurate approximations of the delay probability for small values of s. To test the quality of our approximations, we compare them with the exact value and the Erlang delay formula for some particular cases.