Priority Retrial Research Articles

Lower and upper stochastic bounds for the joint stationary distribution of a non-preemptive priority retrial queueing system

Consider a single-server retrial queueing system with non-preemptive priority service, where customers arrive in a Poisson process with a rate of $\lambda_1$ for high-priority customers (class 1) and $\lambda_2$ for low-priority customers (class 2). If a high-priority customer is blocked, they are queued, while a low-priority customer must leave the service area and return after some random period of time to try again. In contrast with existing literature, we assume different service time distributions for the two customer classes. This investigation proposes a stochastic comparison method based on the general theory of stochastic orders to obtain lower and upper bounds for the joint stationary distribution of the number of customers at departure epochs in the considered model. Specifically, we discuss the stochastic monotonicity of the embedded Markov queue-length process in terms of both the usual stochastic and convex orders. We also perform a numerical sensitivity analysis to study the effect of the arrival rate of high-priority customers on system performance measures.

Cost Optimization of MMAP/PH(1), PH(2)/1 Pre-emptive Priority Retrial Queueing model with Standby Server, Orbital Search, Server Breakdowns, Phase Type repairs and Impatient Customers

Cost Optimization of MMAP/PH(1), PH(2)/1 Pre-emptive Priority Retrial Queueing model with Standby Server, Orbital Search, Server Breakdowns, Phase Type repairs and Impatient Customers

An Analysis of a Two-State Markovian Retrial Queueing Model with Priority Customers

Objective: This study considered a system of retrial queues with two types of customers: high-priority and low-priority. This study deals to find the time dependent probabilities of exact number of arrivals and departures from the system when server is free or busy. Numerical solution and graphical representation will also be presented. Method: For this model, we solved difference differential equations recursively and used Laplace transformation to obtain the transient state probabilities of exact number of arrivals and departures from the system when server is free or busy. Findings: Timedependent probabilities of exact number of arrivals (primary arrivals, arrivals in high priority queue, arrivals in low priority queue) in the system and exact number of departures (primary departures, departures from high priority queue, departures from low priority queue) from the system by a given time for when the server is idle and when the server is busy are obtained. Various interesting performance measures along with some special cases are also obtained. Conversion of two state model into single state model was discussed. Numerical illustrations are also presented using MATLAB programming along with the busy period probabilities of the system and server. Novelty: In past research, models considered arrivals and departures from the orbit whereas in present model arrivals and departures from the system are studied along with the concept of retrial and priority customers. Applications: Priority retrial queues are used in many applications like real time systems, operating systems, manufacturing system, simulation and medical service systems. Keywords: Arrivals; Departures; Probability; Priority; Retrial

Stochastic monotonicity approach for a non-Markovian priority retrial queue

This paper considers a non-Markovian priority retrial queue which serves two types of customers. Customers in the regular queue have priority over the customers in the orbit. This means that the customer in orbit can only start retrying when the regular queue becomes empty. If another customer arrives during a retrial time, this customer is served and the retrial has to start over when the regular queue becomes empty again. In this study, a particular interest is devoted to the stochastic monotonicity approach based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary joint distribution of the embedded Markov chain of the considered system.

Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series approximation method, and the theory of Riemann boundary value problems. We also obtain explicit expressions for the first moments of the stationary distribution under some symmetry assumptions. An application in the modelling of a priority retrial system with coupled orbit queues is also presented. Using a queueing network example, we numerically validated the theoretical findings.

Analysis of non-pre-emptive priority retrial queueing system with two-way communication, Bernoulli vacation, collisions, working breakdown, immediate feedback and reneging

In this study, we investigate a single server priority retrial queueing system including two-way communication, collision, working breakdown, repair, immediate feedback, Bernoulli vacation and reneging. Incoming requests (calls) appear at the service station according to a compound Poisson process. During the idle time, the server can make an outgoing call with an exponentially distributed time. The incoming call that identifies the server occupied will join an orbit or collide with the call currently in service. The server renders the service following a non-pre-emptive priority service rule. The server takes a Bernoulli vacation. The server may become inactive due to normal breakdown and the call currently in service will get the remaining service at a moderate service rate. The repair process starts instantly. After the completion of service, vacation and repair the server is in an idle state. We allow reneging to happen at the orbit. Using the supplementary variable technique, the stability condition is derived.

Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services

In this investigation, a novel sort of retrial queueing system with working breakdown services is introduced. Two distinct kinds of customers are considered, which are priority and ordinary customers. The normal busy server may become inadequate due to catastrophes at any time which cause the major server to fail. At a failure moment, the major server is sent to be fixed and the server functions at a lower speed (called the working breakdown period) during the repair period. The probability generating functions (PGF) of the system size is found using the concepts of the supplementary variable technique (SVT). The impact of parameters in system performance measures and cost optimization are examined numerically.

A study on M/G/1 preemptive priority retrial queue with Bernoulli working vacations and vacation interruption

This paper deals with steady state analysis of single server priority retrial queue with Bernoulli working vacation, where the regular busy server can be subjected to breakdown and repair. There are two types of customers are considered, which are priority customers and ordinary customers. As soon as orbit becomes empty, the server goes for a working vacation (WV). The server works at a lower service rate during working vacation period. If there are customers in the system at the end of each vacation, the server becomes idle and ready for serving new arrivals with probability p (single WV) or it remains on vacation with probability q (multiple WVs). Using the supplementary variable technique, we obtained the steady state probability generating functions for the system and its orbit. Important system performance measures, the mean busy period and the mean busy cycle are discussed. Finally, some numerical examples are presented.

ANALYSIS OF NON PRE EMPTIVE PRIORITY RETRIAL QUEUEING SYSTEM WITH TWO WAY COMMUNICATION, BERNOULLI VACATION, COLLISIONS, WORKING BREAKDOWN, IMMEDIATE FEEDBACK AND RENEGING

In this study, we investigate a single server priority retrial queueing system including two-way communication, collision, working breakdown, repair, immediate feedback, Bernoulli vacation and reneging. Incoming requests (calls) appear at the service station according to a compound Poisson process. During the idle time, the server can make an outgoing call with an exponentially distributed time. The incoming call that identifies the server occupied will join an orbit or collide with the call currently in service. The server renders the service following a non-pre-emptive priority service rule. The server takes a Bernoulli vacation. The server may become inactive due to normal breakdown and the call currently in service will get the remaining service at a moderate service rate. The repair process starts instantly. After the completion of service, vacation and repair the server is in an idle state. We allow reneging to happen at the orbit. Using the supplementary variable technique, the stability condition is derived.

A study on M/G/1 preemptive priority retrial queue with Bernoulli working vacations and vacation interruption

This paper deals with steady state analysis of single server priority retrial queue with Bernoulli working vacation, where the regular busy server can be subjected to breakdown and repair. There are two types of customers are considered, which are priority customers and ordinary customers. As soon as orbit becomes empty, the server goes for a working vacation (WV). The server works at a lower service rate during working vacation period. If there are customers in the system at the end of each vacation, the server becomes idle and ready for serving new arrivals with probability p (single WV) or it remains on vacation with probability q (multiple WVs). Using the supplementary variable technique, we obtained the steady state probability generating functions for the system and its orbit. Important system performance measures, the mean busy period and the mean busy cycle are discussed. Finally, some numerical examples are presented.

Asymptotics of queue length distributions in priority retrial queues

We calculate asymptotics of the distribution of the number of customers in orbit in a two-class priority retrial M∕G∕1-type queueing model. In this model, priority customers wait in line while non-priority customers join an orbit and retry later. Although the generating function and moments of the number of customers in orbit have been analyzed before, asymptotics of the distribution have not been thoroughly investigated, mainly because of the complex nature of the generating function. We show that we can use singularity analysis of the probability generating function to do just that. Our results show that different regimes exist for these asymptotics in case of light-tailed service times: in what we call the ‘priority regime’, the tail asymptotics have the same decay (∼cn−3∕2R−n) as in the priority non-retrial queue and the retrial rate only influences the constant c. In the ‘retrial regime’, the retrial rate also influences the sub-exponential factor of the asymptotics. In this regime, asymptotics are very similar to asymptotics in retrial queues without (priority) waiting line. Finally, we also analyze the case that the service time distribution is power law (with or without exponential cut-off) using the same technique.

Analysis on preemptive priority retrial queue with two types of customers, balking, optional re-service, single vacation and service interruption

This paper concerned with performance analysis of single server preemptive priority retrial queue with a single vacation where two types of customers are considered and they are called priority customers and ordinary customers. The ordinary customers arrive in batch into the system. The priority customers do not form any queue. After the completion of regular service, the customers may demand re-service for the previous service without joining the orbit or may leave the system. As soon as the system is empty, the server goes for vacation and the regular busy server can be subjected to breakdown. By using the supplementary variable technique, we obtain the steady-state probability generating functions for the system/orbit size. Some important system performance measures and the stochastic decomposition are discussed. Finally, numerical examples are presented to visualize the effect of parameters on system performance measures.

Analysis of a preemptive priority retrial queue with negative customers, starting failure and at most J vacations

Analysis of a preemptive priority retrial queue with negative customers, starting failure and at most J vacations

Analysis of a preemptive priority retrial queue with negative customers, starting failure and at most J vacations

An analysis of single server preemptive priority retrial queue with at most J vacations where two types of customers called (priority customers and ordinary customers) are considered in this paper. The priority customers do not have queue and they have higher priority to receive their services over ordinary customers. If negative customer is arriving during the service time of any positive customer (priority customer or ordinary customer), it will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he may join the orbit and the priority customer will leave the system. As soon as the system is empty, the server takes at most J vacations. The probability generating functions for the system/orbit size in steady state is obtained using supplementary variable method. Some important system measures and the stochastic decomposition are discussed. Numerical examples are presented to picturise the effect of parameters on system performance measures.

Performance analysis of preemptive priority retrial queue with immediate Bernoulli feedback under working vacations and vacation interruption

Performance analysis of preemptive priority retrial queue with immediate Bernoulli feedback under working vacations and vacation interruption

Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system

Cognitive radio is emerging as one of the key information transmission technologies to enhance spectrum efficiency for dramatically increased wireless network capacity requested by end users. Dynamic spectrum access allows effective use of radio frequency and prevents its underutilization in many real-world networks. It enables unlicensed users to temporarily “borrow” unused spectrum while ensuring that the rights of the incumbent license holders are respected. Problems of optimization of joint access of primary and secondary users can be effectively solved by means of queueing theory. In this paper, the analysis of a novel queueing model suitable for the optimization of access is implemented under quite general assumptions about the system parameters. There are several types of primary customers having different requirements for the service time and preemptive priority over secondary customers. Secondary customers can share a server, while primary customers occupy the whole server. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of phase-type. Effect of retrials of secondary customers is taken into account. An effective way for the analysis of multi-server queues with many types of customers and heterogeneous requirements to the service process is provided and applied.

Priority retrial queueing model operating in random environment with varying number and reservation of servers

We consider a multi-server queueing system with two types of customers operating in the Markovian random environment. Under the fixed state of the random environment, the arrival flow is described by the marked Markovian arrival process. Type 1 customers have preemptive priority over type 2 customers. Type 1 customer is lost only if, at its arrival moment, all servers are busy by type 1 customers. The service times of both types of customers have an exponential distribution with the rate depending on the type of a customer. Type 2 customer is accepted for service if the number of busy servers at its arrival epoch does not exceed a fixed threshold. Otherwise, the arriving customer makes a randomized choice to leave the system permanently (to balk) or to join so called orbit and try to obtain service later. The inter-retrial times have an exponential distribution. Customers in orbit can be impatient and may leave the system after an exponentially distributed amount of time. Due to preemptive priority of type 1 customers, service of type 2 customer can be interrupted. In this case, the interrupted customer makes a randomized choice to leave the system permanently or go into orbit. When the random environment changes its state, immediately the following parameters of the system change their value: the total number of servers, the number of servers available for type 2 customers, the matrices defining the arrival process, the rates of service of customers, the rate of retrials, the impatience intensity, the probability of balking the system at the arrival moment or the moment of termination of service of type 2 customer. Behavior of the system is described by the level dependent multi-dimensional Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed via the stationary state probabilities. Numerical illustrations are presented.

A preemptive priority retrial queue with two classes of customers and general retrial times

In this paper, we consider a continuous-time retrial queue with two classes of customers: priority customers and ordinary customers, where priority customers don’t queue and have an exclusive preemptive priority to receive their services over ordinary customers. If an arriving ordinary customer finds the server busy, it enters a retrial group (called orbit) according to FCFS discipline. Only the ordinary customer at the head of the retrial queue is allowed to access the server. Firstly, we obtain the necessary and sufficient condition for the system to be stable by embedded Markov chain approach. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures of interest. Thirdly, we give the analysis of the sojourn time in the system of an arbitrary ordinary customer. Lastly, numerical examples are given to show the effect of system parameters on several performance measures.

On the performance of the M1,M2/G1,G2/1 retrial queue with pre-emptive resume policy

Priority mechanism is an invaluable scheduling method that allows customers to receive different quality of service. Service priority is clearly today a main feature of the operation of any manufacturing system. We are interested by an M1,M2/G1,G2/1 priority retrial queue with pre-emptive resume policy. For model in question, we discuss the problem of ergodicity and, by using the method of supplementary variables, find the partial generating functions of the steady state system state distribution. Moreover, some pertinent performance measures are obtained and numerical study is also performed.

Analysis of a Priority Retrial Queue with Dependent Vacation Scheme and Application to Power Saving in Wireless Communication Systems

Analysis of a Priority Retrial Queue with Dependent Vacation Scheme and Application to Power Saving in Wireless Communication Systems