Algorithmic analysis of the multi-server system with a modified Bernoulli vacation schedule

This paper considers an infinite-capacity M/M/c queueing system with modified Bernoulli vacation under a single vacation policy. At each service completion of a server, the server may go for a vacation or may continue to serve the next customer, if any in the queue. The system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition of system equilibrium is obtained. The explicit closed-form of the rate matrix is derived and the useful formula for computing stationary probabilities is developed by using matrix analytic approach. System performance measures are explicitly developed in terms of computable forms. A cost model is derived to determine the optimal values of the number of servers, service rate and vacation rate simultaneously at the minimum total expected cost per unit time. Illustrative numerical examples demonstrate the optimization approach as well as the effect of various parameters on system performance measures.     

有Bernoulli休假和可选服务的M/G/1重试反馈排队模型

考虑具有可选服务的M/G/1重试反馈排队模型,其中服务台有Bernoulli休假策略.系统外新到达的顾客服从参数为λ的泊松过程.重试区域只允许队首顾客重试,重试时间服从一般分布.所有的顾客都必须接受必选服务,然而只有其中部分接受可选服务.每个顾客每次被服务完成后可以离开系统或者返回到重试区域.服务台完成一次服务以后,可以休假也可以继续为顾客服务.通过嵌入马尔可夫链法证明了系统稳态的充要条件.利用补充变量的方法得到了稳态时系统和重试区域中队长分布.我们还得到了重试期间服务台处于空闲的概率,重试区域为空的概率以及其他各种指标.并证出在系统中服务员休假和服务台空闲的时间定义为广义休假情况下也具有随机分解特征.     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

有Bernoulli休假和可选服务的Geo/G/1重试排队

讨论了有Bernoulli休假策略和可选服务的离散时间Geo/G/1重试排队系统.假定一旦顾客发现服务台忙或在休假就进入重试区域,重试时间服从几何分布.顾客在进行第一阶段服务结束后可以离开系统或进一步要求可选服务.服务台在每次服务完毕后,可以进行休假,或者等待服务下一个顾客.还研究了在此模型下的马尔可夫链,并计算了在稳态条件下的系统的各种性能指标以及给出一些特例和系统的随机分解.     

An M[X]/G/1 retrial g-queue with single vacation subject to the server breakdown and repair

An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.     

Geo/Geo/1 retrial queue with non-persistent customers and working vacations

In this paper, we consider a Geo/Geo/1 retrial queue with non-persistent customers and working vacations. The server works at a lower service rate in a working vacation period. Assume that the customers waiting in the orbit request for service with a constant retrial rate, if the arriving retrial customer finds the server busy, the customer will go back to the orbit with probability q (0≤q≤1), or depart from the system immediately with probability $\bar{q}=1-q$ . Based on the necessary and sufficient condition for the system to be stable, we develop the recursive formulae for the stationary distribution by using matrix-geometric solution method. Furthermore, some performance measures of the system are calculated and an average cost function is also given. We finally illustrate the effect of the parameters on the performance measures by some numerical examples.     

有启动失败和可选服务的M/G/1重试排队系统

考虑具有可选服务的M/G/1重试排队模型,其中服务台有可能启动失败.系统外新到达的顾客服从参数为λ的泊松过程.重试区域只允许队首顾客重试,重试时间服务一般分布.所有的顾客都必须接受必选服务,然而只有其中部分接受可选服务.通过嵌入马尔可夫链法证明了系统稳态的充要条件.利用补充变量的方法得到了稳态时系统和重试区域中队长分布.我们还得到重试期间服务台处于空闲的概率,重试区域为空的概率以及其他各种指标.并证出在把系统中服务台空闲和修理的时间定义为广义休假情况下也具有随机分解特征.     

M/(G1, G2)/1 retrial queue under Bernoulli vacation schedules with general repeated attempts and starting failures

This paper investigates a batch arrival retrial queue with general retrial times, where the server is subject to starting failures and provides two phases of heterogeneous service to all customers under Bernoulli vacation schedules. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters a service immediately while the rest join the orbit. After the completion of two phases of service, the server either goes for a vacation with probability p or may wait for serving the next customer with probability (1 − p). We construct the mathematical model and derive the steady-state distribution of the server state and the number of customers in the system/orbit. Such a model has potential application in transfer model of e-mail system.     

Two thresholds of a batch arrival queueing system under modified T vacation policy with startup and closedown

This paper studies the operating characteristics of the variant of an M^[x]/G/1 vacation queue with startup and closedown times. After all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time, and then takes at most J vacations of constant time length T repeatedly until at least one customer is found waiting in the queue upon returning from a vacation. If at least one customer is present in the system when the server returns from a vacation, then the server reactivates and requires a startup time before providing the service. On the other hand, if no customers arrive by the end of the J th vacation, the server remains dormant in the system until at least one customer arrives. We will call the vacation policy modified T vacation policy. We derive the steady‐state probability distribution of the system size and the queue waiting time. Other system characteristics are also investigated. The long‐run average cost function per unit time is developed to determine the suitable thresholds of T and J that yield a minimum cost. Copyright © 2007 John Wiley & Sons, Ltd.     

Batch arrival queues under vacation policies with server breakdowns and startup/closedown times

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under vacation policies with startup/closedown times, where the vacation time, the startup time, and the closedown time are generally distributed. When all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time. After shutdown, the server operates one of (1) multiple vacation policy and (2) single vacation policy. When the server reactivates since shutdown, he needs a startup time before providing the service. If a customer arrives during a closedown time, the service is immediately started without a startup time. The server may break down according to a Poisson process while working and his repair time has a general distribution. We analyze the system characteristics for the vacation models.     

An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule

This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p   (i.e., the vacation is interrupted) or continues the vacation with probability 1-p$1-p$. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented.     

A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair

This paper deals with the steady state behaviour of an M^x/G/1 queue with general retrial time and Bernoulli vacation schedule for an unreliable server, which consists of a breakdown period and delay period. Here we assume that customers arrive according to compound Poisson processes. While the server is working with primary customers, it may breakdown at any instant and server will be down for short interval of time. Further concept of the delay time is also introduced. The primary customer finding the server busy, down or vacation are queued in the orbit in accordance with FCFS (first come first served) retrial policy. After the completion of a service, the server either goes for a vacation of random length with probability p or may continue to serve for the next customer, if any with probability (1 − p). We carry out an extensive analysis of this model. Finally, we obtain some important performance measures and reliability indices of this model.     

A modified vacation model M[x]/G/1 system

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a modified 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. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Further, we derive some important characteristics including the expected length of the busy period and idle period. This shows that the results generalize those of the multiple vacation policy and the single vacation policy M^[x]/G/1 queueing system. Finally, a cost model is developed to determine the optimum of J at a minimum cost. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal management of the machine repair problem with working vacation: Newton’s method

This paper studies the M/M/1 machine repair problem with working vacation in which the server works with different repair rates rather than completely terminating the repair during a vacation period. We assume that the server begins the working vacation when the system is empty. The failure times, repair times, and vacation times are all assumed to be exponentially distributed. We use the MAPLE software to compute steady-state probabilities and several system performance measures. A cost model is derived to determine the optimal values of the number of operating machines and two different repair rates simultaneously, and maintain the system availability at a certain level. We use the direct search method and Newton’s method for unconstrained optimization to repeatedly find the global minimum value until the system availability constraint is satisfied. Some numerical examples are provided to illustrate Newton’s method.     

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.     

Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule

We consider a single server queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under the so called linear retrial policy. This model extends both the classical M/G/1 retrial queue with linear retrial policy as well as the M/G/1 queue with two phases of service and Bernoulli vacation model. We carry out an extensive analysis of the model.     

An M/G/1 Retrial Queueing System with Two-Phase Service and Preemptive Resume

An M/G/1 retrial queueing system with additional phase of service and possible preemptive resume service discipline is considered. For an arbitrarily distributed retrial time distribution, the necessary and sufficient condition for the system stability is obtained, assuming that only the customer at the head of the orbit has priority access to the server. The steady-state distributions of the server state and the number of customers in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analysed numerically. A general decomposition law for this retrial queueing system is established.     

具有不耐烦顾客和PH分布休假的M/M/1排队系统

研究了具有不耐烦顾客的M/M/1休假排队系统,其中休假时间服从位相分布.当顾客在休假时间到达系统,顾客则会因为等待变得不耐烦.服务员休假结束后立刻开始工作.如果在顾客不耐烦时间段内,系统的休假还没有结束,顾客就会离开系统不再回来.建立的模型为水平相依QBD拟生灭过程,通过利用BrightTaylor算法得到系统的稳态概率解.同时还得到一些重要的性能指标.最后通过数据实例验证了我们的结论.     

Performance analysis of a GI/M/1 queue with single working vacation

Consider a GI/M/1 queue with single working vacation. During the vacation period, the server works at a lower rate rather than stopping completely, and only takes one vacation each time. Using the matrix analytic approach, the steady-state distributions of the number of customers in the system at both arrival and arbitrary epochs are obtained. Then the closed property of the conditional probability of gamma distribution is proved and using it the waiting time of an arbitrary customer is analyzed. Finally, Some numerical results and effect of critical model parameters on performance measures have been presented.     

Geo/Geo/1 retrial queue with working vacations and vacation interruption

Consider a Geo/Geo/1 retrial queue with working vacations and vacation interruption, and assume requests in the orbit try to get service from the server with a constant retrial rate. During the working vacation period, customers can be served at a lower rate. If there are customers in the system after a service completion instant, the vacation will be interrupted and the server comes back to the normal working level. We use a quasi birth and death process to describe the considered system and derive a condition for the stability of the model. Using the matrix-analytic method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.