Product form in networks of queues with batch arrivals and batch services

A product form equilibrium distribution is derived for a class of queueing networks in either discrete or continuous time, in which multiple customers arrive simultaneously and batches of customers complete service simultaneously.     

Synchronized queues with deterministic arrivals

We consider m independent exponential servers in parallel, driven by the same deterministic input. This is a modification of the Flatto-Hahn-Wright model which turns out to be easily tractable. We focus on the time-stationary distribution of the number of customers which is obtained using the Palm inversion formula.     

Single-server queues with spatially distributed arrivals

Consider a queueing system where customers arrive at a circle according to a homogeneous Poisson process. After choosing their positions on the circle, according to a uniform distribution, they wait for a single server who travels on the circle. The server's movement is modelled by a Brownian motion with drift. Whenever the server encounters a customer, he stops and serves this customer. The service times are independent, but arbitrarily distributed. The model generalizes the continuous cyclic polling system (the diffusion coefficient of the Brownian motion is zero in this case) and can be interpreted as a continuous version of a Markov polling system. Using Tweedie's lemma for positive recurrence of Markov chains with general state space, we show that the system is stable if and only if the traffic intensity is less than one. Moreover, we derive a stochastic decomposition result which leads to equilibrium equations for the stationary configuration of customers on the circle. Steady-state performance characteristics are determined, in particular the expected number of customers in the system as seen by a travelling server and at an arbitrary point in time.     

Improved bounds for queues with delayed arrivals

Improved bounds are developed for a queue where arrivals are delayed by a fixed time. For moderate to heavy traffic, a simple improved upper bound is obtained which only uses the first two moments of the service time distribution. We show that our approach can be extended to obtain bounds for other types of delayed arrival queues. For very light traffic, asymptotically tight bounds can be obtained using more information about the service time distribution. While an improved upper bound can be obtained for light to moderate traffic it is not particularly easy to apply. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exponential bounds for queues with Markovian arrivals

Exponential bounds [queueb]e ^b are found for queues whose increments are described by Markov Additive Processes. This is done by application of maximal inequalities to exponential martingales for such processes. Through a thermodynamic approach the constant is shown to be the decay rate for an asymptotic lower bound for the queue length distribution. The class of arrival processes considered includes a wide variety of Markovian multiplexer models, and a general treatment of these is given, along with that of Markov modulated arrivals. Particular attention is paid to the calculation of the prefactor .     

Transient analysis of queues with heterogeneous arrivals

In this paper we consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions: (i) an event is either an arrival of typei of batch sizeb _i, i=1,...,r with probability _i or is a depature of a single customer with probability or zero depending on whether the queue is busy or empty; (ii) no more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is 1––_{i i} or 1–_{i i} according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space we will derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.     

OPTIMAL CONTROL FOR A TANDEM NETWORK OF QUEUES WITH BLOCKING

1.IntroductionWeconsideratwo-stationtandemqueuewithnointermediatebuffer.Jobsatthefirststationmaybeblockedwhenthefollowingstationisoccupiedbyanotherjob.Thatis,ajobisblockedatstationoneuponservicecompletionifthefollowingstationisbeingoccupiedbyanotherjob.Afixedcostischargedforeveryenteringjob,forexample,thiscostcanbetheinputrawmaterialcostinamanufacturingsystem.Jobsinthesystemaresubjecttoaholdingcost.Arewardiscollectedwhenajobdepartsfromthesystem(i.e.,fromthesecondstation).Theproblemistocontro…     

On the distributions of infinite server queues with batch arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

     

Nonstationary queues with Interrrupted Poisson arrivals and unreliable/repairable servers

A queueing model having a nonstationary Interrupted Poisson arrival process (IPP(t)),s time-dependent exponential unreliable/repairable servers and finite capacityc is introduced, and an approximation method for analysis of it is developed and tested. Approximations are developed for the time-dependent queue length moments and the system viewpoint waiting time distributions and moments. The approximation involves state-space partitioning and numerically integrating partial-moment differential equations (PMDEs). Surrogate distribution approximations (SDA's) are used to close the system of PMDEs. The approximations allow for analysis using only (s + 1)(s + 6) differential equations for the queue length moments rather than the 2(c + 1)(s +1) equations required by the classic method of numerically integrating the full set of Kolmogorov-forward equations. Effectively hours of cpu time are reduced to minutes for even modest capacity systems. Approximations for waiting time distributions and moments are developed.This research was partially funded by National Science Foundation grant ECS-8404409.     

Algorithmic methods for multi-server queues with group arrivals and exponential services

We discuss algorithms for the computation of the steady-state features of the c-server queue with exponential service times and bounded group arrivals. While our methods are valid for general interarrival time distributions, we treat in detail the simplifications obtained by using a distribution of phase type. Oueue length densities at times prior to arrivals and at arbitrary times are obtained in a modified matrix-geometric form. Their means and variances are found in computationally tractable forms. We also present algorithmic methods for the waiting time and the virtual waiting time distributions of a customer in a group and obtain the means and variances of these distributions in tractable forms. Numerical examples show that the effects of changing various parameters of the queuing model may so be examined at a small computational cost.     

Busy-period and blocking behavior of finite queues with state-dependent Markov renewal arrivals

The busy-period length distributions and blocking probabilities are considered for finiteG/G/1/K queues with state-dependent Markov renewal arrivals. The Laplace-Stieltjes transforms of the distributions and blocking probabilities are given for the non-preemptive and last-come-first-served preemptive resume (or repeat) service disciplines. For Erlangian (or deterministic) service times in particular, it is proved that the busy-period length (the number of blocked customers) for the non-preemptive discipline is smaller (larger) than for the preemptive resume discipline.     

A tandem of discrete-time queues with arrivals and departures at each stage

A discrete-time system of a tandem of queues with exogenous arrivals and departures at each stage is considered. A customer leaving queuek–1 departs the system with probability 1– ^[k] and continues to queuek with probability ^[k] . Exogenous arrivals to each stage are i.i.d. at each time slot. An approximate analysis of the occupancy and busy-period distributions of each stage based on a General Busy-period with batches and Memoryless (geometric) Idle period renewal Process (GBMIP) provides improved performance over two-state Markov approximations and gives exact results when there are no interstage departures.This research was supported in part by NSF grant NCR-8708282.     

Numerical analysis of multi-server queues with deterministic service and special phase-type arrivals

In this paper we study multi-server queues with deterministic service times and a phase-type arrival process. In particular, we consider for the interarrival time hyperexponential distributions and mixtures of Erlang distributions with the same scale parameter. We give an algorithm to compute the steady state probabilities of the number of customers in the system and derive explicit expressions for various operating characteristics.Special attention is paid to the numerical evaluation of the required transition probabilities and the implementation of the iterative solution method for the steady state probabilities.

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Zusammenfassung In dieser Arbeit werden Bedienungssysteme mit mehreren parallelen Schaltern studiert, wobei ein Ankunftsstrom vom Phasentyp und deterministische Bedienungszeiten angenommen werden. Speziell werden hyperexponentialverteilte Zwischenankunftszeiten betrachtet oder solche, die nach einer Mischung von Erlangverteilungen mit dem selben Skalenparameter verteilt sind. Es werden ein Algorithmus zur Berechnung der stationären Wahrscheinlichkeiten für die Anzahl der Kunden im System angegeben sowie explizite Ausdrücke für diverse Charakteristiken hergeleitet.Besonderes Gewicht liegt auf der numerischen Berechnung der benötigten Übergangswahrscheinlichkeiten und der Implementierung der iterativen Methode zur Berechnung der stationären Wahrscheinlichkeiten.

     

Regularly varying tails in a queue with discrete autoregressive arrivals of order p

We consider a discrete time single server queueing system where the service time of a customer is one slot, and the arrival process is governed by a discrete autoregressive process of order p (DAR(p)). For this queueing system, we investigate the tail behavior of the queue size and the waiting time distributions. Specifically, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β−1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

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.     

Two competing discrete-time queues with priority

This paper considers a class of two discrete-time queues with infinite buffers that compete for a single server. Tasks requiring a deterministic amount of service time, arrive randomly to the queues and have to be served by the server. One of the queues has priority over the other in the sense that it always attempts to get the server, while the other queue attempts only randomly according to a rule that depends on how long the task at the head of the queue has been waiting in that position. The class considered is characterized by the fact that if both queues compete and attempt to get the server simultaneously, then they both fail and the server remains idle for a deterministic amount of time. For this class we derive the steady-state joint generating function of the state probabilities. The queueing system considered exhibits interesting behavior, as we demonstrate by an example.     

Threshold policies for controlled retrial queues with heterogeneous servers

Retrial queues are an important stochastic model for many telecommunication systems. In order to construct competitive networks it is necessary to investigate ways for optimal control. This paper considers K -server retrial systems with arrivals governed by Neut' Markovian arrival process, and heterogeneous service time distributions of general phase-type. We show that the optimal policy which minimizes the number of customers in the system is of a threshold type with threshold levels depending on the states of the arrival and service processes. An algorithm for the numerical evaluation of an optimal control is proposed on the basis of Howar's iteration algorithm. Finally, some numerical results will be given in order to illustrate the system dynamics. AMS subject classification: 60K25 93E20     

On M/M/1 queues with a smart machine

This paper discusses a class of M/M/1 queueing models in which the service time of a customer depends on the number of customers served in the current busy period. It is particularly suited for applications in which the server has kind of learning ability and warms up gradually. We present a simple and computationally tractable scheme which recursively determines the stationary probabilities of the queue length. Other performance measures such as the Laplace transform of the busy period are also obtained. For the firstN exceptional services model which can be considered as a special case of our model, we derive a closed-formula for the generating function of the stationary queue length distribution. Numerical examples are also provided.     

Regeneration and renovation in queues

Regenerative events for different queueing models are considered. The aim of this paper is to construct these events for continuous-time processes if they are given for the corresponding discrete-time model. The construction uses so-called renovative events revealing the property of the state at timen of the discrete-time model to be independent (in an algebraic sense) of the states referring to epochs not later thann −L (whereL is some constant) given that there are some restrictions on the “governing sequence”. Different types of multi-server and multi-phase queues are considered.