The Embedded Random Walk in the Stationary M/M/1 Queue

We consider the stationary queue length process of the standard M/M/1 queue just after its jump times (i.e., the times of arrivals or departures). Simple formulas for the distribution of this embedded random walk with immediate reflection at zero are derived. We study the monotonicity properties of the corresponding expected values and point out an asymmetry between the number of arrivals and the number of departures until the n-th jump time.     

On bulk-service MAP/PH L,N /1/N G-Queues with repeated attempts

We consider a retrial queue with a finite buffer of size N, with arrivals of ordinary units and of negative units (which cancel one ordinary unit), both assumed to be Markovian arrival processes. The service requirements are of phase type. In addition, a PH^L,N bulk service discipline is assumed. This means that the units are served in groups of size at least L, where 1≤ L≤ N. If at the completion of a service fewer than L units are present at the buffer, the server switches off and waits until the buffer length reaches the threshold L. Then it switches on and initiates service for such a group of units. On the contrary, if at the completion of a service L or more units are present at the buffer, all units enter service as a group. Units arriving when the buffer is full are not lost, but they join a group of unsatisfied units called “orbit”. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a level dependent quasi-birth-and-death process. We start by analyzing a simplified version of our queueing model, which is amenable to numerical calculation and is based on spatially homogeneous quasi-birth-and-death processes. This leads to modified matrix-geometric formulas that reveal the basic qualitative properties of our algorithmic approach for computing performance measures. AMS Subject Classification: Primary 60K25 Secondary 68M20 90B22.     

Analyzing the discrete-timeG (G)/Geo/1 queue using complex contour integration

In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG ^(G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG ^(G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.Both authors wish to thank the Belgian National Fund for Scientific Research (NFWO) for support of this work.     

An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

Asymptotic Results for Ruin Probability of a Two-Dimensional Renewal Risk Model

In this article, some asymptotic formulas of the finite-time ruin probability for a two-dimensional renewal risk model are obtained. In the model, the distributions of two claim amounts belong to the intersection of the long-tailed distributions class and the dominated varying distributions class and the claim arrival-times are extended negatively dependence structures. Assumption that the claim arrivals of two classes are governed by a common renewal counting process. The asymptotic formulas hold uniformly for t ∈ [f(x), ∞), where f(x) is an infinitely increasing function.     

Departures inGR^{X{n}} /G_n /\infty

Departure processes in infinite server queues with non-Poisson arrivals have not received much attention in the past. In this paper, we try to fill this gap by considering the continuous time departure process in a general infinite server system with a Markov renewal batch arrival process ofM different types of customers. By a conditional approach, analytical results are obtained for the generating functions and binomial moments of the departure process. Special cases are discussed, showing that while Poisson arrival processes generate Poisson departures, departure processes are much more complicated with non-Poisson arrivals.This research has been supported in part by the Natural Science and Engineering Research Council of Canada (Grant A5639).     

On the steady-state queue size distribution of the discrete-timeGeo/G/1 queue with repeated customers

In this paper, we study the steady-state queue size distribution of the discrete-timeGeo/G/1 retrial queue. We derive analytic formulas for the probability generating function of the number of customers in the system in steady-state. It is shown that the stochastic decomposition law holds for theGeo/G/1 retrial queue. Recursive formulas for the steady-state probabilities are developed. Computations based on these recursive formulas are numerically stable because the recursions involve only nonnegative terms. Since the regularGeo/G/1 queue is a special case of theGeo/G/1 retrial queue, the recursive formulas can also be used to compute the steady-state queue size distribution of the regularGeo/G/1 queue. Furthermore, it is shown that a continuous-timeM/G/1 retrial queue can be approximated by a discrete-timeGeo/G/1 retrial queue by dividing the time into small intervals of equal length and the approximation approaches the exact when the length of the interval tends to zero. This relationship allows us to apply the recursive formulas derived in this paper to compute the approximate steady-state queue size distribution of the continuous-timeM/G/1 retrial queue and the regularM/G/1 queue.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0046415.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0105828.     

The Busy Period and the Waiting Time Analysis of a MAP/M/c Queue with Finite Retrial Group

Abstract

We concentrate on the analysis of the busy period and the waiting time distribution of a multi-server retrial queue in which primary arrivals occur according to a Markovian arrival process (MAP). Since the study of a model with an infinite retrial group seems intractable, we deal with a system having a finite buffer for the retrial group. The system is analyzed in steady state by deriving expressions for (a) the Laplace–Stieltjes transforms of the busy period and the waiting time; (b) the probabiliy generating functions for the number of customers served during a busy period and the number of retrials made by a customer; and (c) various moments of quantites of interest. Some illustrative numerical examples are discussed.     

Markov renewal queues of M/M(a,d)/l/(b,n) system-part i

This paper discusses a general bulk service queue which falls into the Markov renewal class. Applying an analysis similar to the one by Hunter (1983) for M/M1/N type of feedback queues, certain properties of discrete and continuous time queue length processe are studied here. The results and formulas are then applied to a numerical illustration.     

TheGr X n/G n/∞ system: System size

An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.     

Some exact expressions for the bulk-arrival queue mx/M/1

For a M/M/1 queueing system with group arrivals of random size the transition probabilities of the queue size process and the distribution of the maximal queue size during a time interval [0,t) are calculated. Simple formulae for the corresponding Laplace transforms are given.     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

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.     

On the service system MX/G/∞

In the present paper we consider the service system M^X/G/∞ characterized by an infinite number of servers anda general service time distribution. The customers arrive at the system in groups of size X, which is a random variable, the time between group arrivals being exponentially distributed. Using simple probability arguments, we obtain probability generating functions (p.g.f.'s) of the number of busy servers at time t and the number that depart by time t. Several other properties of these random variables are also discussed.     

Delay analysis of two batch-service queueing models with batch arrivals: Geo X /Geo c /1

In this paper, we compute the probability generating functions (PGF’s) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF’s, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches.     

<Emphasis Type="Italic">n</Emphasis>-Colour even self-inverse compositions

An n-colour even self-inverse composition is defined as an n-colour selfinverse composition with even parts. In this paper, we get generating functions, explicit formulas and recurrence formulas for n-colour even self-inverse compositions. One new binomial identity is also obtained.     

Analysis of the M/G/1 processor-sharing queue with bulk arrivals

We analyze the single server processor-sharing queue for the case of bulk arrivals. We obtain an expression for the expected response time of a job as a function of its size, when the service times of jobs have a generalized hyperexponential distribution and more generally for distributions with rational Laplace transforms. Our analysis significantly extends the class of distributions for which processor-sharing queues with bulk arrivals were previously analyzed.     

Performance analysis of discrete-time queue GI/G/1 with negative arrivals

In this paper, we consider a discrete-time GI/G/1 queueing model with negative arrivals. By deriving the probability generating function of actual service time of ordinary customers, we reduced the analysis to an equivalent discrete-time GI/G/1 queueing model without negative arrival, and obtained the probability generating function of buffer contents and random customer delay.     

The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue

This paper studies a single server queueing system with multiple types of customers. The first part of the paper discusses some modeling issues associated with the Markov arrival processes with marked arrivals (MMAP[K], where K is an integer representing the number of types of customers). The usefulness of MMAP[K] in modeling point processes is shown by a number of interesting examples. The second part of the paper studies a single server queueing system with an MMAP[K] as its input process. The busy period, virtual waiting time, and actual waiting times are studied. The focus is on the actual waiting times of individual types of customers. Explicit formulas are obtained for the Laplace–Stieltjes transforms of these actual waiting times.