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

In this paper, we analyze a single-server vacation queue with a general arrival process. Two policies, working vacation and vacation interruption, are connected to model some practical problems. The GI/M/1 queue with such two policies is described and by the matrix analysis method, we obtain various performance measures such as mean queue length and waiting time. Finally, using some numerical examples, we present the parameter effect on the performance measures and establish the cost and profit functions to analyze the optimal service rate η during the vacation period.     

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.     

在D-策略控制下服务员单重休假且休假不中断的M/G/1排队系统分析

该文研究在D-策略控制下服务员单重休假且休假不中断的M/G/1排队系统,其中当服务员休假结束归来时,如果系统中等待服务的顾客所需的总服务时间之和不小于事先给定的正数阀值D,服务员就立即开始服务.运用全概率分解技术、更新过程理论和拉普拉斯变换工具,本文在任意初始状态下讨论了队长的瞬态分布,导出了队长瞬态分布的拉普拉斯变换的表达式和稳态队长分布的递推表达式.同时给出了稳态队长的随机分解结构、附加队长分布的显示表达式.进一步借用稳态队长分布{pj,j=0,1,2,?},讨论了系统容量的优化设计,并阐述了稳态队长分布对系统容量优化设计所起的重要作用.最后,在建立费用模型的基础上,导出了系统在长期单位时间内期望费用的显示表达式,并通过数值实例不仅确定了使系统在长期单位时间内的期望费用最小的控制策略D?,而且还得到了当休假时间长度为固定时长T(>0)时系统的联合控制策略(T?,D?).     

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.     

带负顾客和N-策略的Geo^{lambda_1, lambda_2/Geo/1(MWV)排队系统分析及最优控制策略N*

分析了一个带有负顾客、N-策略控制的Geo/Geo/1多重工作休假排队系统, 其中正顾客在工作休假及正规忙期以不同的到达率进入系统. 利用拟生灭过程和矩阵几何解方法, 给出了该模型的稳态队长分布及平均队长, 以及系统分别处于假期和忙期的概率. 同时, 对该系统的忙期进行了分析, 并讨论了稳态队长分布在系统容量的优化设计中的应用. 最后, 在给定的费用结构下, 用数值计算例子确定了使系统长期单位时间内期望费用最小的最优控制策 N*.     

A heuristic algorithm for the optimization of a retrial system with Bernoulli vacation

In this study, we consider an M/M/c retrial queue with Bernoulli vacation under a single vacation policy. When an arrived customer finds a free server, the customer receives the service immediately; otherwise the customer would enter into an orbit. After the server completes the service, the server may go on a vacation or become idle (waiting for the next arriving, retrying customer). The retrial system is analysed as a quasi-birth-and-death process. The sufficient and necessary condition of system equilibrium is obtained. The formulae for computing the rate matrix and stationary probabilities are derived. The explicit close forms for system performance measures are developed. A cost model is constructed to determine the optimal values of the number of servers, service rate, and vacation rate for minimizing the total expected cost per unit time. Numerical examples are given to demonstrate this optimization approach. The effects of various parameters in the cost model on system performance are investigated.     

Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation

This paper analyzes the F-policy M/M/1/K queueing system with working vacation and an exponential startup time. The F-policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K, the optimal threshold F and the optimal service rates (μ_B,μ_V) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.     

延迟控制休假的M／M／1排队系统

本文研究带有延迟休假的 Ｍ／Ｍ／１排队系统,服务员在空闲了一段时间（称做延迟时间）后才正式开始休假,每次休假的时间长度有指数分布．若一次休假结束时系统中的顾客数目低于某一水平Ｋ,则服务员开始另一次休假;否则转为投入服务,这时系统开始一个新的忙期。对于延迟时间有指数分布和是确定的情形分别求得系统的稳态分布的精确表示及某些性能指标．文章还讨论了系统优化问题,给出使得单位时间平均总成本最小的Ｋ值．证明在泊松到达的情形最优延迟时间是０（无延迟）或无穷（无休假）     

On M/G/1 system under NT policies with breakdowns,startup and closedown

This paper studies the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times, in which the length of the vacation period is controlled either by the number of arrivals during the vacation period, or by a timer. After all the customers are served in the queue exhaustively, the server is shutdown (deactivates) by a closedown time. At the end of the shutdown time, the server immediately takes a vacation and operates two different policies: (i) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the closedown time. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. If some customers arrive during this closedown time, the service is immediately started without leaving for a vacation and without a startup time. We analyze the system characteristics for each scheme.     

带有部分工作休假和休假中断的M/M/c排队

考虑了一个带有部分工作休假和休假中断的多服务台M/M/c排队.在休假期,d(d相似文献   

不同价格水平下排队策略研究

在M/M/1排队中引入了不同的服务价格,基于"收益-成本"结构,以顾客和企业均追求利益最大化为出发点,在两种不可见情形下,研究了顾客均衡策略行为和企业最优服务定价决策,通过数值模拟,描述了休假期服务价格对顾客均衡策略的影响,以及几乎不可见情况下休假期服务价格对企业收益的作用和完全不可见情况下休假期服务价格随潜在到达率的变化情况,以及当企业获得最大收益时,正常工作期和休假期服务价格的关系.     

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.     

带有负顾客的M/M/c多重工作休假排队

考虑服务台在休假期间不是完全停止工作,而是以相对于正常服务期低些的服务率服务顾客的M/M/c工作休假排队模型.在此模型基础上,针对现实的M/M/c排队模型中可能出现的外来干扰因素,提出了带有负顾客的M/M/c工作休假排队这一新的模型.服务规则为先到先服务.工作休假策略为空竭服务异步多重工作休假.抵消原则为负顾客一对一抵消处于正常服务期的正顾客,若系统中无处于正常服务期的正顾客时,到达的负顾客自动消失,负顾客不接受服务.首先,由该多重休假模型得到其拟生灭过程及生成元矩阵,然后运用矩阵几何方法给出系统队长的稳态分布表达式和若干系统指标.     

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.     

Optimality of a Threshold Policy in theM/M/ queue with repeated vacations

Consider anM/M/1 queueing system with server vacations where the server is turned off as soon as the queue gets empty. We assume that the vacation durations form a sequence of i.i.d. random variables with exponential distribution. At the end of a vacation period, the server may either be turned on if the queue is non empty or take another vacation. The following costs are incurred: a holding cost ofh per unit of time and per customer in the system and a fixed cost of each time the server is turned on. We show that there exists a threshold policy that minimizes the long-run average cost criterion. The approach we use was first proposed in Blanc et al. (1990) and enables us to determine explicitly the optimal threshold and the optimal long-run average cost in terms of the model parameters.     

Notes of M/G/1 system under the {\langle p, T \rangle} policy with second optional service

We optimize the operating cost of the ${\langle p, T \rangle}We optimize the operating cost of the áp, T ?{\langle p, T \rangle} policy for an M/G/1 queueing system with second optional service, where the customer may depart from the system either after the first essential service with probability 1 − r or at the end of the first service may immediately go for a second service with probability r. Moreover, the server takes a vacation of fixed length T if the system becomes empty. If customers are found in the queue after T time units have elapsed since the end of the busy period, the server reactivates with probability p or leaves for a vacation of the same length T with probability 1 − p. Alternatively, if no customers present in the queue upon returning from the vacation, the server leaves for another a vacation of the same length. We call this áp, T ?{\langle p, T \rangle} policy. The total expected cost function per unit time is developed to determine the optimal thresholds of p and T at a minimum cost. Based on the optimal cost the explicit form for joint optimum values of p and T are obtained.     

The discrete-time MAP/PH/1 queue with multiple working vacations

We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed.     

The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0 ? p ? 1) or continue the vacation with probability 1 − p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.     

A discrete queue with double thresholds policy and its application to SVC systems

A discrete time Geo/Geo/1 queue with (m, N)-policy is considered in this paper. There are three operation periods being considered: high speed, low speed service periods and idle periods. With double thresholds policy, the server begins to take a working vacation when the number of customers is below m after a service and there is one customer in the system at least. What’s more, if the system becomes empty after a service, the server will take an ordinary vacation. Otherwise, high speed service continues if the number of customers still exceeds m after a service. At the vacation completion instant, servers resume their service if the quantity of customers exceeds N. Vacations can also be interrupted when the system accumulate customers more than the prefixed threshold. Using the quasi birth-death process and matrix-geometric solution methods, we derive the stationary queue length distribution and some system characteristics of interest. Based on these, we apply the queue to a virtual channel switching system and present various numerical experiments for the system. Finally, numerical results are offered to illustrate the optimal (m, N)-policy to minimize cost function and obtain practical consequence on the operation of double thresholds policy.     

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.