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.     

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.     

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.     

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.     

Optimality of D-Policies for an M/G/1 Queue with a Removable Server

We consider an M/G/1 queue with a removable server. When a customer arrives, the workload becomes known. The cost structure consists of switching costs, running costs, and holding costs per unit time which is a nonnegative nondecreasing right-continuous function of a current workload in the system. We prove an old conjecture that D-policies are optimal for the average cost per unit time criterion. It means that for this criterion there is an optimal policy that either runs the server all the time or switches the server off when the system becomes empty and switches it on when the workload reaches or exceeds some threshold D.     

The M/G/c queue in light traffic

Under light traffic, we investigate the quality of a well‐known approximation for first‐moment performance measures for an M/G/c queue, and, in particular, conditions under which the approximation is either an upper or a lower bound. The approach is to combine known relationships between quantities such as average delay and time‐average work in system with direct sample‐path comparisons of system operation under two modes of operation: conventional FIFO and a version of preemptive LIFO. We then use light traffic limit theorems to show an inequality between time‐average work of the M/G/c queue and that of the approximation. In the process, we obtain new and improved approximations. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

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.     

Consecutive customer losses in regular and oscillating <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">n</Emphasis> systems

We derive fast recursions to compute the probability that k or more consecutive customer losses take place during a busy period of a queue, the so called k-CCL probability, for regular and oscillating M ^X /G/1/n systems.     

Time-dependent analysis of M/G/1 vacation models with exhaustive service

We analyze the time-dependent process in severalM/G/1 vacation models, and explicitly obtain the Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state. Exhaustive-serviceM/G/1 systems with multiple vacations, single vacations, an exceptional service time for the first customer in each busy period, and a combination ofN-policy and setup times are considered. The decomposition property in the steady-state joint distribution of the queue size and the remaining service time is demonstrated.     

Analysis of the M/M/1 Queue with Processor Sharing via Spectral Theory

We show in this paper that the computation of the distribution of the sojourn time of an arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS queue) can be formulated as a spectral problem for a self-adjoint operator. This approach allows us to improve the existing results for this queue in two directions. First, the orthogonal structure underlying the M/M/1 PS queue is revealed. Second, an integral representation of the distribution of the sojourn time of a customer entering the system while there are n customers in service is obtained.     

Connection admission control for constant bit rate traffic at a multi‐buffer multiplexer using the oldest‐cell‐first discipline

Consider an ATM multiplexer where M input links contend for time slots on an output link which transmits C cells per second. Each input link has its own queue of size B cells. The traffic is delay sensitive so B is small (e.g., B=20). We assume that each of the M input links carries Constant Bit Rate (CBR) traffic from a large number of independent Virtual Connections (VCs) which are subject to jitter. The fluctuations of the aggregate traffic arriving at queue i, i=1,...,M, is modeled by a Poisson process with rate λ_i. The Quality of Service (QoS) of one connection is determined in part by the queueing delay across the multiplexer and the Cell Loss Ratio (CLR) or proportion of cells from this connection lost because the buffer is full. The Oldest‐Customer(Cell)‐First (OCF) discipline is a good compromise between competing protocols like round‐robin queueing or serving the longest queue. The OCF discipline minimizes the total cell delay among all cells arriving at the contending queues. Moreover, the CLR is similar to that obtained by serving the longest queue. We develop QoS formulae for this protocol that can be calculated on‐line for Connection Admission Control (CAC). These formulae follow from a simple new expression for the exact asymptotics of a M/D/1 queue. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue

In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.     

Busy period analysis for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">PH</Emphasis>/1 queues with workload dependent balking

We consider an M/PH/1 queue with workload-dependent balking. An arriving customer joins the queue and stays until served if and only if the system workload is no more than a fixed level at the time of his arrival. We begin by considering a fluid model where the buffer content changes at a rate determined by an external stochastic process with finite state space. We derive systems of first-order linear differential equations for the mean and LST (Laplace-Stieltjes Transform) of the busy period in this model and solve them explicitly. We obtain the mean and LST of the busy period in the M/PH/1 queue with workload-dependent balking as a special limiting case of this fluid model. We illustrate the results with numerical examples.      

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

On the multi-phase M/G/1 queueing system with random feedback

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ _i the (i + 1)-th phase starts, with probability p _i the customer feedback to the tail of the queue and with probability 1 − θ _i − p _i  = q _i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p _k feedback to the tail of the queue and with probability 1 − p _k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.     

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM _t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.     

Analysis of M/G/1-Queues with Setup Times and Vacations under Six Different Service Disciplines

Single server M/G/1-queues with an infinite buffer are studied; these permit inclusion of server vacations and setup times. A service discipline determines the numbers of customers served in one cycle, that is, the time span between two vacation endings. Six service disciplines are investigated: the gated, limited, binomial, exhaustive, decrementing, and Bernoulli service disciplines. The performance of the system depends on three essential measures: the customer waiting time, the queue length, and the cycle duration. For each of the six service disciplines the distribution as well as the first and second moment of these three performance measures are computed. The results permit a detailed discussion of how the expected value of the performance measures depends on the arrival rate, the customer service time, the vacation time, and the setup time. Moreover, the six service disciplines are compared with respect to the first moments of the performance measures.     

The stationary workload of the <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 queue with impatient customers

We consider a G/M/1 queue in which the patience time of the customers is constant. The stationary distribution of the workload of the server, or the virtual waiting time, is derived by the level crossing argument. To this end, we obtain the expected downcrossings of a level in the workload process during a busy cycle and then the expected length of a busy cycle. For both the expectations, we use the dual property between the M/G/1 and G/M/1 queue.