Stability of the multiserver queue with addressed retrials

In the recent paper (Mushko et al. in Ann. Oper. Res. 141:283??301, 2006) Mushko, Jacob, et al. considered an M/M/c type queueing system with retrials. Given that returning customers have access to any server they obtained a sufficient condition for the stability of the system. We suggest an alternative approach to the problem and get the necessary and sufficient condition for the stability in more general situation, when some servers are reserved for processing of primary requests and do not serve returning customers.     

AnM/M/1 retrial queue with control policy and general retrial times

We consider anM/M/1 retrial queueing system in which the retrial time has a general distribution and only the customer at the head of the queue is allowed to retry for service. We find a necessary and sufficient condition for ergodicity and, when this is satisfied, the generating function of the distribution of the number of customers in the queue and the Laplace transform of the waiting time distribution under steady-state conditions. The results agree with known results for special cases.Supported by KOSEF 90-08-00-02.     

Mobile customer model with retrials

A cellular system consisting of small zones is studied. Since their zones are small, the change of the number of mobile customers in a cell influences the performance. The hand-off failure probability and blocking probability may be important as the performance measures. In this paper, we consider the retrial behavior of customers who meet the hand-off failure and blocking. We classify customers into three types: the retrial resignation type, the ordinary retrial type and the persistent retrial type. We evaluate the effect of the existence of mobile customers with retrials.     

Approximation of M/M/c retrial queue with PH-retrial times

We consider the M/M/c retrial queues with PH-retrial times. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. Some numerical results are presented.     

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.     

On the distribution of the number of retrials

We consider queuing systems where customers are not allowed to queue, instead of that they make repeated attempts, or retrials, in order to enter service after some time. We obtain the distribution of the number of retrials produced by a tagged customer, until he finds an available server.     

An M/M/1 retrial queue with unreliable server

We analyze an unreliable M/M/1 retrial queue with infinite-capacity orbit and normal queue. Retrial customers do not rejoin the normal queue but repeatedly attempt to access the server at i.i.d. intervals until it is found functioning and idle. We provide stability conditions as well as several stochastic decomposability results.     

Threshold control by a single-server retrial queue with batch arrivals and group services

A controlled single-server retrial queueing system is investigated. Customers arrive according to batch Markovian arrival process. The system has several operation modes which are controlled by means of a threshold strategy. The stationary distribution is calculated. Optimization problem is considered and a numerical example is presented.     

A retrial queue with structured batch arrivals,priorities and server vacations

A retrial queue accepting two types of customers with correlated batch arrivals and preemptive resume priorities is studied. The service times are arbitrarily distributed with a different distribution for each type of customer and the server takes a single vacation each time he becomes free. For such a model the state probabilities are obtained both in a transient and in a steady state. Finally, the virtual waiting time of an arbitrary ordinary customer in a steady state is analysed.     

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM _t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.     

Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.     

The Israeli queue with retrials

The so-called “Israeli queue” (Boxma et al. in Stoch Model 24(4):604–625, 2008; Perel and Yechiali in Probab Eng Inf Sci, 2013; Perel and Yechiali in Stoch Model 29(3):353–379, 2013) is a multi-queue polling-type system with a single server. Service is given in batches, where the batch sizes are unlimited and the service time of a batch does not depend on its size. After completing service, the next queue to be visited by the server is the one with the most senior customer. In this paper, we study the Israeli queue with retrials, where the system is comprised of a “main” queue and an orbit queue. The main queue consists of at most \(M\) groups, where a new arrival enters the main queue either by joining one of the existing groups, or by creating a new group. If an arrival cannot join one of the groups in the main queue, he goes to a retrial (orbit) queue. The orbit queue dispatches orbiting customers back to the main queue at a constant rate. We analyze the system via both probability generating functions and matrix geometric methods, and calculate analytically various performance measures and present numerical results.     

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.     

The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.     

Analysis of an retrial queue

We consider a single server retrial queueing system in which each customer (primary or retrial customer) has discrete service times taking on value D_j with probability , and . An arriving primary customer who finds the server busy tries later. Moreover, each retrial customer has its own orbit, and the retrial customers try to enter the service independently of each other. We call this retrial queue an M/{D_n}/1 retrial queue. A necessary and sufficient condition for this system stability is given. In the steady state, we derive the joint distribution of the state of the server and the number of customers in the retrial orbits. The explicit expressions of some performance measures are given. In addition, the steady-state distribution of the waiting time is discussed.     

A repairable Geo X /G/1 retrial queue with Bernoulli feedback and impatient customers

Abstract This paper deals with a discrete-time batch arrival retrial queue with the server subject to starting failures.Diferent from standard batch arrival retrial queues with starting failures,we assume that each customer after service either immediately returns to the orbit for another service with probabilityθor leaves the system forever with probability 1θ(0≤θ1).On the other hand,if the server is started unsuccessfully by a customer(external or repeated),the server is sent to repair immediately and the customer either joins the orbit with probability q or leaves the system forever with probability 1 q(0≤q1).Firstly,we introduce an embedded Markov chain and obtain the necessary and sufcient condition for ergodicity of this embedded Markov chain.Secondly,we derive the steady-state joint distribution of the server state and the number of customers in the system/orbit at arbitrary time.We also derive a stochastic decomposition law.In the special case of individual arrivals,we develop recursive formulae for calculating the steady-state distribution of the orbit size.Besides,we investigate the relation between our discrete-time system and its continuous counterpart.Finally,some numerical examples show the influence of the parameters on the mean orbit size.     

Single line queue with recurrent repeated demands

We analyze a model queueing system in which customers cannot be in continuous contact with the server, but must call in to request service. If the server is free, the customer enters service immediately, but if the server is occupied, the unsatisfied customer must break contact and reapply for service later. There are two types of customer present who may reapply. First transit customers who arrive from outside according to a Poisson process and if they find the server busy they join a source of unsatisfied customers, called the orbit, who according to an exponential distribution reapply for service till they find the server free and leave the system on completion of service. Secondly there are a number of recurrent customers present who reapply for service according to a different exponential distribution and immediately go back in to the orbit after each completion of service. We assume a general service time distribution and calculate several characterstic quantities of the system for both the constant rate of reapplying for service and for the case when customers are discouraged and reduce their rate of demand as more customers join the orbit.     

批到达M/G/1重试排队的队长的尾渐近

用随机分解法研究成批到达服务时间为次指数分布的重试排队中队长的尾行为,得到了该系统与其相应的标准排队系统队长尾分布的关系;对次指数尾,结果也能用于正则变化尾,进而得到正则变化尾渐近．     

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 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.