Assia Boumahdaf
We consider an $M/M/1$ queueing system with impatient customers with multiple and single vacations. It is assumed that customers are impatient whenever the state of the server. We derive the probability generating functions of the number of customers in the system and we obtain some performance measures.
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In this paper we models and studies a general vacation queueing model with impatient customers. We first propose a sufficient condition for the existence of the stationary workload process. We then give an integral equation for the independent and identically distributed case. This integral equation is solved when customers arrive according to a Poisson Point process. A relationship between the tail of the waiting time distribution and the tail of service distribution is also given.
Vacation queueing systems are widely used as an extension of the classical queueing theory. We consider both working vacations and regular vacations in this paper, and compare systems with vacations to the regular $M/M/1$ system via mean service rates and expected numbers of customers, using matrix-analytic methods.
We investigate a processor sharing queue with renewal arrivals and generally distributed service times. Impatient jobs may abandon the queue, or renege, before completing service. The corresponding stochastic processes are represented by measure valued Markov processes on R^2\_+. A scaling procedure that gives rise to a fluid model with a nontrivial, yet tractable steady state behavior, is presented. This fluid model model captures many essential features of the underlying stochastic model, and it is used to analyze the impact of impatience in processor sharing queues.
An infinite buffer batch service vacation queue has been studied where service rate of the batch is dependent on the size of the batch and vacation rate is dependent on the queue size at vacation initiation epoch. The arrivals follow the Markovian arrival process (MAP). For service rule, general bulk service (GBS) rule is considered. The service time and vacation time both are considered to be generally distributed. Several joint distributions of interest are obtained using the bivariate vector generating function method and the supplementary variable technique (SVT). Numerical results are pre…
In this paper, we deal with an $D/GI/1$ vacation system with impatient customers. We give a sufficient condition for the existence of a limit distribution of the waiting time and integral equations are derived in both reneging and balking scenarios. Explicit solutions are given when vacation times are exponentially distributed and service times are either exponentially distributed or deterministic.
We present the explicit construction of a stable queue with several servers and impatient customers, under stationary ergodic assumptions. Using a stochastic comparison of the (multivariate) workload sequence with two monotonic stochastic recursions, we propose a sufficient condition of existence of a unique stationary state of the system using Renovation theory. Whenever this condition is relaxed we use extension techniques to prove the existence of a stationary state in some cases.
The Gated-Service M/GI/1 Queue with Single Vacations and Its Application to Batch-Service Queues
本文研究了具有单个休假的Gated服务M/GI/1队列，推导了稳态队列长度的概率生成函数、系统延迟分布的Laplace-Stieltjes变换以及忙周期长度和服务顾客数的联合变换。同时，将结果应用于批量服务M/G/1队列，其中服务时间依赖于批量中的顾客数。
In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N -(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relat…
We introduce a queueing system that alternates between two modes, so-called {\it working mode} and {\it vacation mode}. During the working mode the system runs as an $M^{X}/G/1$ queue. Once the number of customers in the working mode drops to zero the vacation mode begins. %Then working system becomes empty the vacation phase begins. During the vacation mode the system runs as a general queueing system (a service might be included) which is different from the one in the working mode. The vacation period ends in accordance with a given stopping rule, and then a random number of customers are tr…
This paper studies the equilibrium behavior of customers in the Geo/Geo/1 queueing system with multiple working vacations. The arriving customers decide whether to join or to balk the queueing systems based on the information of the queue length and the states of the server. In observable queues, partially observable queues and unobservable queues three cases we obtain the equilibrium balking strategies based on the reward-cost structure and socially optimal strategies for all customers. Furthermore, we present some numerical experiments to illustrate the effect of the information level on the…
In this paper, we consider a Lévy-driven fluid queueing system where the server may subject to breakdowns and repairs. In addition, the server will leave for a vacation each time when he finds an empty system. We cast the queueing process as a Lévy process modified to have random jumps at two classes of stopping times. By using the Kella-Whitt martingale method, we obtain the limiting distribution of the virtual waiting time process. Moreover, we investigate the busy period, the correlation structure and the stochastic decomposition properties. These results may be generalized to Lévy processe…
Discrete-time queueing system has widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the probabilities in terms of …
This paper focuses on the problem of modeling the correspondence pattern for ordinary people. Suppose that letters arrive at a rate $λ$ and are answered at a rate $μ$. Furthermore, we assume that, for a constant $T$, a letter is disregarded when its waiting time exceeds $T$, and the remains are answered in {\it last in first out} order. Let $W_n$ be the waiting time of the $n$-th {\it answered} letter. It is proved that $W_n$ converges weekly to $W_T$, a non-negative random variable which possesses a density with {\it power-law} tail when $λ=μ$ and with exponential tail otherwise. Note that th…
We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacatio…
We consider a broad class of queueing models with random state-dependent vacation periods, which arise in the analysis of queue-based back-off algorithms in wireless random-access networks. In contrast to conventional models, the vacation periods may be initiated after each service completion, and can be randomly terminated with certain probabilities that depend on the queue length. We examine the scaled queue length and delay in a heavy-traffic regime, and demonstrate a sharp trichotomy, depending on how the activation rate and vacation probability behave as function of the queue length. In p…
In this paper, equilibrium strategies and optimal balking strategies of customers in a constant retrial queue with multiple vacations and the $N$-policy under two information levels, respectively, are investigated. We assume that there is no waiting area in front of the server and an arriving customer is served immediately if the server is idle; otherwise (the server is either busy or on a vacation) it has to leave the system to join a virtual retrial orbit waiting for retrials according to the FCFS rules. After a service completion, if the system is not empty, the server becomes idle, availab…
A two-class Processor-Sharing queue with one impatient class is studied. Local exponential decay rates for its stationary distribution (N, M) are established in the heavy traffic regime where the arrival rate of impatient customers grows proportionally to a large factor A. This regime is characterized by two time-scales, so that no general Large Deviations result is applicable. In the framework of singular perturbation methods, we instead assume that an asymptotic expansion of the solution of associated Kolmogorov equations exists for large A and derive it in the form P(N = Ax, M = Ay) ~ g(x,y…
We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.
In this paper we study a two-queue polling model with zero switch-over times and $k$-limited service (serve at most $k_i$ customers during one visit period to queue $i$, $i=1,2$) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as th…
Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacation where customers/messages arrive according to the Bernoulii process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at arbitrary slot…
Queueing-inventory Systems: A Survey
本文系统综述了2016至2023年排队-库存系统的理论与应用进展，涵盖乘积形式解、易腐库存、重试休假、多服务器、博弈论分析及能量收集与供应链等应用，并指出挑战与未来方向。
We consider an infinite-buffer single-server queue where inter-arrival times are phase-type ($PH$), the service is provided according to Markovian service process $(MSP)$, and the server may take single, exponentially distributed vacations when the queue is empty. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function (VGF) of system-length distribution at a pre-arrival epoch. Also, we obtain the steady-state system-length distribution at an arbitrary epoch along with some important performance measures such as the mean number of cus…
Queue occupancy and server size distribution of a queue length dependent vacation queue with an optional service
本文研究了具有依赖队列长度的休假队列，推导了双变量概率生成函数，并通过数值示例展示了框架的应用性及参数敏感性分析。
We study a many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales, so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin-Whitt regime, (b) the nondegenerate slowdown regime, and (c) the interme…
The PS-model treated in this paper is motivated by freelance job websites where multiple freelancers compete for a single job. In the context of such websites, multistage service of a job means collection of applications from multiple freelancers. Under Markovian stochastic assumptions, we develop fluid limit approximations for the PS-model in overload. Based on this approximation, we estimate what proportion of freelancers get the jobs they apply for. In addition, the PS-model studied here is an instant of PS with routing and impatience, for which no Lyapunov function is known, and we suggest…
We study a double-ended queue which consists of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system immediately. The matching follows first-come-first-serve principle. If a customer from one class cannot be matched immediately, he/she will stay in a queue and wait for the upcoming arrivals from the other class. Thus there cannot be non-zero numbers of customers from both classes simultaneously in the system. We also assume that each customer can leave the queue without being matched because of impatience. The arrival processe…
This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information they decide to wait for service or to leave the system. Specifically, every customer has a patience threshold and they balk if the observed delay is above the threshold. The main objective is to estimate the parameters of the customers' patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication, and distinguishing feature of our setup, lies in th…
We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idli…
In this research, we study the information freshness in M/G/1 queueing system with a single buffer and the server taking multiple vacations. This system has wide applications in communication systems. We aim to evaluate the information freshness in this system with both i.i.d. and non-i.i.d. vacations under three different scheduling policies, namely Conventional Buffer System (CBS), Buffer Relaxation System (BRS), and Conventional Buffer System with Preemption in Service (CBS-P). For the systems with i.i.d. vacations, we derive the closed-form expressions of information freshness metrics such…
The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the $M/SM/1$ queue, has been well-studied since its introduction by Neuts in 1966. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic. In particular, we use it in the context of gap acceptance models where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority. Traditionally, gap acceptance…
In the study of large scale stochastic networks with resource management, differential equations and mean-field limits are two key techniques. Recent research shows that the expected fraction vector (that is, the tailed probability vector) plays a key role in setting up mean-field differential equations. To further apply the technique of tailed probability vector to deal with resource management of large scale stochastic networks, this paper discusses tailed probabilities in some basic queueing processes including QBD processes, Markov chains of GI/M/1 type and of M/G/1 type, and also provides…
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