A single server queue with random arrivals and balking: Confidence interval estimation

This paper investigates a queueing system, which consists of Poisson input of customers, some of whom are lost to balking, and a single server working a shift of lengthL and providing a service whose duration can vary from customer to customer. If a service is in progress at the end of a shift, the server works overtime to complete the service. This process was motivated by the behavior of fishermen interviewed in the NY Great Lakes Creel Survey.We derive the distributions of the number of services (X), overtime and total server idle time (T), both unconditionally (for Poisson arrivals) and conditionally on the number (n) of arrivals per shift, assuming that the arrival times are not recorded in the data. These distributions provide the basis for estimation of the parameters from asingle realization of the queueing process during [0,L]. The conditional distributions also can be used to estimate common service time,w, when (n, X) or (n, T) are observed. Confidence intervals based onT are of shorter length, for all confidence coefficients, than the corresponding intervals based onX.This paper is Technical Report #BU-1019-M in the Biometrics Unit Series. The authors are grateful to N.U. Prabhu for suggestions on streamlining the distributional derivations and to D.R. Cox and C.E. McCulloch for helpful comments.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A single server model for packetwise transmission of messages

We study a discrete-time single-server queue where batches of messages arrive. Each message consists of a geometrically distributed number of packets which do not arrive at the same instant and which require a time unit as service time. We consider the cases of constant spacing and geometrically distributed (random) spacing between consecutive packets of a message. For the probability generating function of the stationary distribution of the embedded Markov chain we derive in both cases a functional equation which involves a boundary function. The stationary mean number of packets in the system can be computed via this boundary function without solving the functional equation. In case of constant (random) spacing the boundary function can be determined by solving a finite-dimensional (an infinite-dimensional) system of linear equations numerically. For Poisson- and Bernoulli-distributed arrivals of messages numerical results are presented. Further, limiting results are derived.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

The equivalence of the transient behaviour formulae for the single server queue

It is indeed obvious to expect that the different results obtained for some problem are equal, but it needs to established. For the M/M/1/N queue, using a simple algebraic approach we will prove that the results obtained by Takâcs [17] Takâcs, L. 1960. Introduction to the Theory of Queues Oxford University Press.  [Google Scholar] and Sharma and Gupta [13] Sharma, O.P. and Gupta, U.C. 1982. Transient Behaviour of an M/M/l/N Queue. Stoch. Process Appl., 13: 327–331. [Crossref] , [Google Scholar] are equal. Furthermore, a direct proof to the equivalence between all formulae of the M/M/l/∞ queue is established. At the end of this paper, we will show that the time-dependent state probabilities for M/M/l/N queue can be written in series form; its coefficients satisfy simple recurrence relations which would allow for the rapid and efficient evaluation of the state probabilities. Moreover, a brief comparison of our technique, Sharma and Gupta's formula and Takâcs result is also given, for the CPU time computing the state probabilities.     

Logarithmic asymptotics for a single-server processing distinguishable sources

We consider a single-server first-in-first-out queue fed by a finite number of distinct sources of jobs. For a large class of short-range dependent and light-tailed distributed job processes, using functional large deviation techniques we prove a large deviation principle and logarithmic asymptotics for the joint waiting time and queue lengths distribution. We identify the paths that are most likely to lead to the rare events of large waiting times and long queue lengths. A number of examples are presented to illustrate salient features of the results.      

Performance evaluation of a single queue capable of handling two like jobs as a single entity

In this paper, we model an open queuing network and analyze performance measures with and without feedback as two individual cases. The model comprises a single queue with a dedicated processor capable of handling two like jobs as a single job. Two different job arrivals with different processing times are considered with an internal timer. Performance measures such as average queue length, average response time, average waiting time of the jobs are computed and plotted. The joint density function for the inter arrival time and arrival rate are derived. The probability mass function has been derived for all possible cases that may arise in a duration (0, t], considering n job arrivals during that period of time and an integer programming problem is formulated to obtain optimal sequence patterns which would maximize the efficiency of the model.     

On the infinite server shortest queue problem: Non-symmetric case

We consider two parallel M/M/∞ queues. All servers in the first queue work at rate μ₁ and all in the second work at rate μ₂. A new arrival is routed to the system with the lesser number of customers. If both queues have equal occupancy, the arrival joins the first queue with probability ν₁, and the second with probability ν₂ = 1−ν₁. We analyze this model asymptotically. We assume that the arrival rate λ is large compared to the two service rates. We give several different asymptotic formulas, that apply for different ranges of the state space. The numerical accuracy of the asymptotic results is tested. AMS subject classification 60K25 60K30 34E20     

The queue-length distribution for Mx/G1 queue with single server vacation

1 IntroductionDuring recent decades many authors studied M/G/l queues with server vacations (seeRefS[1 ~ 6]). They not only studied the stocliastic decomposition properties of the queue lengthand waiting time when the system is in equilibrium, but also studied its transient and equilibrium distributions. Although Baba[7] studied bulk-arrival M"/G/1 with vacation time andShils] studied a kind of M"/G(M/H)/1 queue with repairable service station, they didll't studythe transient and equilibr…     

Large deviation principle of stochastic differential equations with non-Lipschitzian coefficients

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang＇s results.     

A large-deviations analysis of the GI/GI/1 SRPT queue

We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT results to compare SRPT with FIFO from a large-deviations point of view. 2000 Mathematics Subject Classification: Primary—60K25; Secondary—60F10; 90B22     

GI／G／1排队系统的队长和等待时间的瞬时分布

本是[1，2]的继续,在本中利用马氏骨架过程给出了GI／G／1排队系统的队长的瞬时分布的另一新的计算方法和等待时间的计算方法。     

An approximate analysis of a cyclic server queue with limited service and reservations

In this paper, we examine a queueing problem motivated by the pipeline polling protocol in satellite communications. The model is an extension of the cyclic queueing system withM-limited service. In this service mechanism, each queue, after receiving service on cyclej, makes a reservation for its service requirement in cyclej + 1. The main contribution to queueing theory is that we propose an approximation for the queue length and sojourn-time distributions for this discipline. Most approximate studies on cyclic queues, which have been considered before, examine the means only. Our method is an iterative one, which we prove to be convergent by using stochastic dominance arguments. We examine the performance of our algorithm by comparing it to simulations and show that the results are very good.     

Path Space Large Deviations of a Large Buffer with Gaussian Input Traffic

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having different Hurst indices.     

Large deviation for stochastic Cahn-Hilliard partial differential equations

In this paper, we prove a large deviation principle for a class of stochastic Cahn-Hilliard partial differential equations driven by space-time white noises.     

推广的单重休假M~x/G/1排队系统

研究了服务前需要重新调整机器的单重休假Mx/G/1排队系统,在LS变换和L变换下得到了服务员忙期中队长的瞬态分布和队长稳态分布的概率母函数.     

Large deviation principle for reflected Poisson driven stochastic differential equations in epidemic models

Abstract

We establish a large deviation principle for a reflected Poisson driven stochastic differential equation. Our motivation is to study in a forthcoming paper the problem of exit of such a process from the basin of attraction of a locally stable equilibrium associated with its law of large numbers. Two examples are described in which we verify the assumptions that we make to establish the large deviation principle.     

Large sample inference from single server queues

Problems of large sample estimation and tests for the parameters in a single server queue are discussed. The service time and the interarrivai time densities are assumed to belong to (positive) exponential families. The queueing system is observed over a continuous time interval (0,T] whereT is determined by a suitable stopping rule. The limit distributions of the estimates are obtained in a unified setting, and without imposing the ergodicity condition on the queue length process. Generalized linear models, in particular, log-linear models are considered when several independent queues are observed. The mean service times and the mean interarrival times after appropriate transformations are assumed to satisfy a linear model involving unknown parameters of interest, and known covariates. These models enhance the scope and the usefulness of the standard queueing systems.Partially supported by the U. S. Army Research Office through the Mathematical Sciences Institute of Cornell University.