Fluid polling systems

We study N-queues single-server fluid polling systems, where a fluid is continuously flowing into the queues at queue-dependent rates. When visiting and serving a queue, the server reduces the amount of fluid in the queue at a queue-dependent rate. Switching from queue i to queue j requires two random-duration steps: (i) departing queue i, and (ii) reaching queue j. The length of time the server resides in a queue depends on the service regime. We consider three main regimes: Exhaustive, Gated, and Globally-Gated. Two polling procedures are analyzed: (i) cyclic and (ii) probabilistic. Under steady-state, we derive the Laplace–Stieltjes transform (LST), mean, and second moment of the amount of flow at each queue at polling instants, as well as at an arbitrary moment. We further calculate the LST and mean of the “waiting time” of a drop at each queue and derive expressions for the mean total load in the system for the various service regimes. Finally, we explore optimal switching procedures.     

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.     

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 a GI/M/1 queue with multiple working vacations

Consider a GI/M/1 queue with vacations such that the server works with different rates rather than completely stops during a vacation period. We derive the steady-state distributions for the number of customers in the system both at arrival and arbitrary epochs, and for the sojourn time for an arbitrary customer.     

On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation

We consider finite buffer single server GI/M/1 queue with exhaustive service discipline and multiple working vacations. Service times during a service period, service times during a vacation period and vacation times are exponentially distributed random variables. System size distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system etc. have been obtained. The model has potential application in the area of communication network, computer systems etc. where a single channel is allotted for more than one source.     

The Analysis of a General Input Queue with N Policy and Exponential Vacations

This paper studies a single removable server in a G/M/1 queueing system with finite capacity where the server applies an N policy and takes multiple vacations when the system is empty. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential and deterministic interarrival time distributions. We establish the distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs, as well as the distributions of the waiting time and the busy period.     

Performance Analysis of the GI/M/1 Queue with Single Working Vacation and Vacations

In this paper, we consider a new class of the GI/M/1 queue with single working vacation and vacations. When the system become empty at the end of each regular service period, the server first enters a working vacation during which the server continues to serve the possible arriving customers with a slower rate, after that, the server may resume to the regular service rate if there are customers left in the system, or enter a vacation during which the server stops the service completely if the system is empty. Using matrix geometric solution method, we derive the stationary distribution of the system size at arrival epochs. The stochastic decompositions of system size and conditional system size given that the server is in the regular service period are also obtained. Moreover, using the method of semi-Markov process (SMP), we gain the stationary distribution of system size at arbitrary epochs. We acquire the waiting time and sojourn time of an arbitrary customer by the first-passage time analysis. Furthermore, we analyze the busy period by the theory of limiting theorem of alternative renewal process. Finally, some numerical results are presented.     

推广的多重休假$M^X/G/1$排队系统

在平稳状态下,Baba利用补充变量方法研究了多重休假的MX/G/1排队,但作者假定了休假时间和服务时间都有概率密度函数.本文考虑推广的多重休假MX/G/1排队,在假定休假时间和服务时间都是一般概率分布函数下,我们研究了队长的瞬态和稳态性质.通过引进"服务员忙期"和使用不同于Baba文中使用的分析技术,我们导出了在任意时刻t瞬态队长分布的L变换的递推表达式和稳态队长分布的递推表达式,以及平稳队长的随机分解.特别地,通过本文可直接获得多重休假的M/G/1与标准的MX/G/1排队系统相应的结果.     

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time.     

The GI/M/1 queue with phase-type working vacations and vacation interruption

Consider a GI/M/1 queue with phase-type working vacations and vacation interruption where the vacation time follows a phase-type distribution. The server takes the original work at the lower rate during the vacation period. And, the server can come back to the normal working level at a service completion instant if there are customers at this instant, and not accomplish a complete vacation. From the PH renewal process theory, we obtain the transition probability matrix. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at arrival epochs, and waiting time of an arbitrary customer. Meanwhile, we obtain the stochastic decomposition structures of the queue length and waiting time. Two numerical examples are presented lastly.     

Performance analysis of M/G/1 queue with working vacations and vacation interruption

In this paper, an M/G/1 queue with a working vacations and vacation interruption is analyzed. Using the method of a supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at an arbitrary epoch under steady state conditions. Further, we provide the Laplace-Stieltjes transform (LST) of the stationary waiting time. Finally, numerical examples are presented.     

具有一般休假的D-BMAP/G/1分解定理

为了直接求出队长向量母函数,省去繁琐的矩阵分析或补充变量过程,给出并证明了具有一般休假的D—BMAP/G/1的分解定理,阐明了任意时刻队长的向量母函数与闲期任意时刻队长向量母函数的关系,离去时刻队长的向量母函数与闲期任意时刻队长向量母函数的关系,闲期任意时刻队长向量母函数与Y_(busy)(θ,z),即忙期任意时刻队长和剩余服务时间的联合向量母函数的关系．     

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).     

Analysis of a Two‐Phase Queueing System with Vacations and Bernoulli Feedback

Abstract

We consider a two‐phase queueing system with server vacations and Bernoulli feedback. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. Each customer who completes the individual service returns to the tail of the second phase service queue with probability 1 ? σ. If the system becomes empty at the moment of the completion of the second phase services, the server takes vacations until he finds customers. This type of queueing problem can be easily found in computer and telecommunication systems. By deriving a relationship between the generating functions for the system size at various embedded epochs, we obtain the system size distribution at an arbitrary time. The exhaustive and gated cases for the batch service are considered.     

The performance measures and randomized optimization for an unreliable server M/G/1 vacation system

This paper examines an M^[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 − p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. Whenever one or more customers arrive at server idle state, the server immediately starts providing service for the arrivals. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the distributions of important system characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, the distributions of idle period, busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗, J^∗) at a minimum cost, and some numerical examples are presented for illustrative purpose.     

Operating characteristic analysis on the M/G/1 system with a variant vacation policy and balking

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. If the server is busy or on vacation, an arriving batch balks (refuses to join) the system with probability 1 − b. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are derived along with some numerical illustration.     

Analysis of the MAP/G/1/N queue with multiple vacations

This paper deals with a batch service queue and multiple vacations. The system consists of a single server and a waiting room of finite capacity. Arrival of customers follows a Markovian arrival process (MAP). The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’. We obtain the queue length distributions at various epochs along with some key performance measures. Finally, some numerical results have been presented.     

A Recursive Algorithm for State Dependent GI/M/1/N Queue with Bernoulli-Schedule Vacation

In this paper, we study a renewal input working vacations queue with state dependent services and Bernoulli-schedule vacations. The model is analyzed with single and multiple working vacations. The server goes for exponential working vacation whenever the queue is empty and the vacation rate is state dependent. At the instant of a service completion, the vacation is interrupted and the server resumes a regular busy period with probability 1???q (if there are customers in the queue), or continues the vacation with probability q (0?≤?q?≤?1). We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, using some numerical results, we present the parameter effect on the various performance measures.     

服务员假期中以概率p进入的M/G/1排队系统的随机分解

该文研究M/G/1多重休假排队系统，其中在服务员休假中到达顾客以概率p(0≤p≤1)进入。通过引进“服务员忙期”和使用拉普拉斯变换或拉普拉斯— —司梯阶变换，我们获得队长瞬态分布的拉普拉斯变换和稳态分布的递推表达式，进一步得到稳态队长分布的随机分解和在特殊情况下相应的一些结果。     

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.