Analysis of discrete-time queues with batch renewal input and multiple vacations

This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed.Service and vacation times are mutually independent and geometrically distributed.The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch.The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system.Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival,arbitrary and outside observer’s observation epochs for partial-batch rejection policy.The blocking probability of the first-,an arbitrary-and the last-job in a batch have been discussed.The analysis of actual waiting-time distributions measured in slots of the first-,an arbitrary- and the last-job in an accepted batch,and other performance measures along with some numerical results have also been investigated.     

具有多重休假和Min(N,V)-策略控制的Geo/G/1离散时间排队

考虑服务员具有多重休假和系统采用min(N,V)-策略控制的离散时间Geo/G/1排队系统,使用全概率分解技术和更新过程理论,研究了系统在任意时刻n+的瞬态队长分布和稳态队长分布,得到了瞬态队长分布的z-变换表达式和稳态队长分布的递推表达式.进一步,得到了系统在时刻点n,n~-和外部观察时刻点的稳态队长分布.特别地,本文直接获得了一些特殊离散时间排队系统相应的结果.最后,通过数值实例阐述了获得便于计算的稳态队长分布的表达式在系统容量设计中的重要价值.     

Bernoulli反馈排队的N策略Geom／G／1排队系统的队长分布

考虑具有Bernoulli反馈排队和N策略体假的离散时间Geom/G/1排队系统,采用一种直观、简洁的全概率分解方法以及U-变换技术,研究了该排队系统队长的瞬态性质,得到队长瞬态分布的U-变换形式的递推表达式;进一步导出队长稳态分布的递推表达以及平稳队长分布的随机分解;最后,计算出稳态队长分布,并且考察了其统计性质.     

Queue Size Distribution of Geo/G/1 Queue Under the Min(N,D)-Policy

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D,whichever occurs first(Min(N,D)-policy).By using renewal process theory and total probability decomposition technique,the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state,and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n~+.Meanwhile,the authors obtain the explicit expressions of the additional queue length distribution.Furthermore,the important relations between the steady state queue length distributions at different time epochs n~-,n and n~+ are also reported.Finally,the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution,and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.     

Recursive Solution of Queue Length Distribution for Geo/G/1 Queue with Delayed Min(N,D)-Policy

In this paper we consider a discrete-time Geo/G/1 queue with delayed Min(N, D)-policy.Using renewal process theory, total probability decomposition technique and z-transform, we study the transient and equilibrium properties of the queue length from an arbitrary initial state, and obtain both the recursive expressions of the transient state queue length distribution and the steady state queue length distribution at arbitrary time epoch n~+. Furthermore, we derive the important relations between equilibrium queue length distributions at different time epochs n~-, n and n~+. Finally, we give some numerical examples about capacity decision in queueing systems to demonstrate the application of the analytical results reported in this paper.     

The recursive solution of queue length for <Emphasis Type="Italic">Geo/G</Emphasis>/1 queue with <Emphasis Type="Italic">N</Emphasis>-policy

This paper considers a discrete-time queue with N-policy and LAS-DA(late arrival system with delayed access) discipline.By using renewal process theory and probability decomposition techniques,the authors derive the recursive expressions of the queue-length distributions at epochs n~-,n~+,and n.Furthermore,the authors obtain the stochastic decomposition of the queue length and the relations between the equilibrium distributions of the queue length at different epochs(n~-,n~+,n and departure epoch D_n).     

The M/PH/1 queue with working vacations and vacation interruption

We study an M/PH/1 queue with phase type working vacation and vacation interruption where the vacation time follows a phase type distribution. The server serves the customers at a lower rate in a vacation period. The server comes back to the regular busy period at a service completion without completing the vacation. Such policy is called vacation interruption. In terms of quasi birth and death process and matrix-geometric solution method, we obtain the stationary queue length distribution. Moreover we obtain the conditional stochastic decomposition structures of queue length and waiting time when the service time distribution in the regular busy period is exponential.     

Queue size distribution of Geo/G/1 queue under the Min(<Emphasis Type="Italic">N,D</Emphasis>)-policy

This paper considers a discrete-time Geo/G/1 queue under the Min(N,D)-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first (Min(N,D)-policy). By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of the transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n ⁺. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution. Furthermore, the important relations between the steady state queue length distributions at different time epochs n ^-, n and n ⁺ are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.     

ANALYSIS AND COMPUTATIONAL ALGORITHM FOR QUEUES WITH STATE-DEPENDENT VACATIONS Ⅱ:M(n)/G/1/K

We study a single-server queueing system with state-dependent arrivals and general service-distribution.or simply M(n)/G/1/K.where the server follows an N policy and takes multiple vacationswhen the system is empty.We provide a recursive algorithm using the supplementary variable tech-nique to munerically compute the stationary queue length distribution of the system.The only inputrequirements are the Laplace-Stieltjes transforms of the service time distribution and the vacation timedistribution.and the state-dependent arrival rate.The computational complexity of the algorithm isO(K~3).     

Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations

This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.     

一类休假的Geo~X/G/1可修排队的可靠性分析

考虑单重休假、Bernoulli反馈和可变输入率的离散时间Geo~X/G/1可修排队.顾客的批到达速率与服务器的休假有关.刚服务完的顾客以概率1-θ进入队列寻求下次服务.服务器在服务过程中可能故障需修复后再继续工作.借助更新过程理论、z变换和一种分解法,研究了时刻n+位于服务器忙期的条件概率、服务器的瞬态和稳态不可用度以及(0~+,n~+]时间内服务器的平均故障次数和稳态故障频度,揭示了这类离散时间可修排队中服务器可靠性指标的结构,得到了一些特殊可修排队的可靠性结果.最后通过数值实例分析了系统参数对服务器可靠性指标的影响.     

Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times, and queue lengths

This paper studies a continuous time queueing system with multiple types of customers and a first-come-first-served service discipline.Customers arrive according to a semi-Markov arrival process and the service times of individual types of customers have PH-distributions.A GI/M/1 type Markov process for a generalized age process of batches of customers is constructed.The stationary distribution of the GI/M/1 type Markov process is found explicitly and,consequently,the distributions of the age of the batch in service,the total workload in the system,waiting times,and sojourn times of different batches and different types of customers are obtained.The paper gives the matrix representations of the PH-distributions of waiting times and sojourn times.Some results are obtained for the distributions of queue lengths at departure epochs and at an arbitrary time.These results can be used to analyze not only the queue length,but also the composition of the queue.Computational methods are developed for calculating steady state distributions related to the queue lengths,sojourn times,and waiting times.     

Analysis of batch arrival queue with randomized vacation policy and an un-reliable server

This paper examines an M^[x]G/1 queueing system with an unreliable server and a delayed repair,in which the server operates a randomized vacation policy with multiple 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.Whenever one or more customers arrive when the server is idle,the server immediately starts providing service for the arrivals.The server may also meet an unpredictable breakdown and the repair may be delayed.For such a system the authors derive the distributions of some important system characteristics,such as the system size distribution at a random epoch and at a departure epoch,the system size distribution at the busy period initiation epoch,and the distribution of the idle period and the busy period.The authors perform a numerical analysis for changes in the system characteristics,along with changes in specific values of the system parameters.A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.     

具有负顾客到达和RCH移除策略的GI/D-MSP/1/N离散时间排队系统

综合运用补充变量方法和基于条件概率矩阵迭代的嵌入Markov链方法研究了具有负顾客到达和RCH移除策略的离散时间GI/D-MSP/1/N排队系统. 获得了稳态情形下正顾客到达前夕, 任意时隙分点以及外部观测时刻的三种队长分布. 并进一步讨论了可入系统正顾客的等待时间分布. 最后通过几个特殊情形下的数值算例验证了计算方法理论分析的正确性.     

ANALYSIS AND COMPUTATIONAL ALGORITHM FOR QUEUES WITH STATE-DEPENDENT VACATIONS II： M（n）/G/1/K

We study a single-server queueing system with state-dependent arrivals and general service distribution, or simply M（n）/G/1/K, where the server follows an N policy and takes multiple vacations when the system is empty. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. The only input requirements are the Laplace-Stieltjes transforms of the service time distribution and the vacation time distribution, and the state-dependent arrival rate. The computational complexity of the algorithm is O（K^3）.     

Equilibrium Customer Strategies in Markovian Queues with Setup Times and Partial Failures

This paper considers the equilibrium behavior of customers in a Markovian queue with setup times and partial failures, where the reactivated server must go through a period of setup time to reach the normal working state and a failure can occur at any time during the normal service. When a partial failure occurs, the server continues to serve the customers on spot at a low rate and does not admit a new arrival. Once the system becomes empty, an exponential repair time starts. Assuming that all the customers have the option of joining or balking based on a linear reward-cost structure, the authors analyze the equilibrium strategies of the customers and the average social benefits of the system in the fully observable case and the partially observable case, respectively. And on this basis, the effect of several parameters on customers' strategic behavior is presented by some numerical examples.     

延迟N-策略M/G/1排队系统队长的瞬态和稳态分布

研究延迟N-策略M/G/1排队系统,讨论了队长的瞬态和稳态性质.通过引进"服务员忙期"和使用全概率分解技术,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.     

Randomized policy of a poisson input queue with <Emphasis Type="Italic">J</Emphasis> vacations

This paper studies the operating characteristics of an M/G/1 queuing system with a randomized control policy and at most J vacations.After all the customers are served in the queue exhaustively, the server immediately takes at most J vacations repeatedly until at least N customers are waiting for service in the queue upon returning from a vacation.If the number of arrivals does not reach N by the end of the J~(th) vacation,the server remains idle in the system until the number of arrivals in the queue re...     

A batch arrival retrial queue with starting failures,feedback and admission control

This paper is concerned with the analysis of a feedback M^[X]/G/1 retrial queue with starting failures and general retrial times. In a batch, each individual customer is subject to a control admission policy upon arrival. If the server is idle, one of the customers admitted to the system may start its service and the rest joins the retrial group, whereas all the admitted customers go to the retrial group when the server is unavailable upon arrival. An arriving customer (primary or retrial) must turn-on the server, which takes negligible time. If the server is started successfully (with a certain probability), the customer gets service immediately. Otherwise, the repair for the server commences immediately and the customer must leave for the orbit and make a retrial at a later time. It is assumed that the customers who find the server unavailable are queued in the orbit in accordance with an FCFS discipline and only the customer at the head of the queue is allowed for access to the server. The Markov chain underlying the considered queueing system is studied and the necessary and sufficient condition for the system to be stable is presented. Explicit formulae for the stationary distribution and some performance measures of the system in steady-state are obtained. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.     

An M/M/1 Queueing-Inventory System with Geometric Batch Demands and Lost Sales

This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples.