Orbit Queue Research Articles

Some reflected autoregressive processes with dependencies

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered.

Bounds and Maxima for the Workload in a Multiclass Orbit Queue

In this research, a single-server M-class retrial queueing system (orbit queue) with constant retrial rates and Poisson inputs is considered. The main purpose is to construct the upper and lower bounds of the stationary workload in this system expressed via the stationary workloads in the classical M/G/1 systems where the service time has M-component mixture distributions. This analysis is applied to establish the extreme behaviour of stationary workload in the retrial system with Pareto service-time distributions for all classes.

Energy harvesting cognitive radio networks with strategic users: A two-class queueing model with retrials

Motivated by the rapid growth in wireless communication systems, we develop a single-server queueing model with service retrials for the cognitive radio network (CRN). A key appeal of CRN is its flexible allocation scheme of limited spectrum for two user groups: (1) the licensed user, hereby referred to as primary user (PU), and (2) unlicensed users, referred to as secondary users (SUs). The PU take priority over SUs: when a PU arrives and observes a busy server currently occupied by an SU, it will take over the server by preempting the SU out of service. SUs are strategic and delay sensitive: they decide immediately upon arrivals on whether to join the CRN or to balk, based on the anticipated expected utility (service reward minus delay cost). In case the server is unavailable, SUs temporarily join an orbit queue where they wait for future accesses to service. Another realistic feature of our model is that, after successfully serving a user (PU or SU), the server becomes unavailable and undergoes an energy harvesting phase of a random time. We characterize the equilibrium system performance in the following three steps: First, we obtain the equilibrium joining behavior for utility-maximizing SUs; second, we derive the optimal joining strategy from the perspective of a social planner; and finally, we enforce that users adopt the socially optimal strategy by imposing an admission fee. We also investigate the impact of the energy harvesting time on users?? equilibrium strategies and report numerical experiments to provide qualitative and quantitative insights of our results.

Increasing social welfare with delays: Strategic customers in the M/G/1 orbit queue

Strategic customers typically patronize service systems at a higher rate than the socially optimal one. Much literature has focused on inducing customers to join such systems at this latter rate. This entire literature considers nonidling policies that are the focus of queueing theory. We demonstrate that strategically imposing delays into service systems can improve the social welfare using the M/G/1 queue with orbit. This versatile queueing model has been extensively studied from a performance evaluation perspective. In this system, customers who arrive and find the server idle begin immediately their service. However, strategic customers who find the server busy decide whether to balk or join a virtual queue, that is, an orbit. Then, each time the server finishes a service, he begins to retrieve a customer from the orbit and the corresponding retrieving time is not negligible. These retrieving times function as extra delays that are imposed on customers that find a busy server. We show that when customers are strategic, there are certain ranges of the parameters where delaying the orbit customers can increase the welfare of a system or even maximize it. To this end, we characterize and compute the equilibrium strategies for the customers' joining/balking dilemma. We consider both the unobservable and observable versions of the system, and provide some insight on the optimal delay and level of information in such systems. We further show that the welfare for this system is higher than the corresponding standard M/G/1 queue with the same delay.

Dynamic Rebalancing Strategy in Free-Float Bicycle Sharing Systems: Orbit Queues and Two-Sided Matching

In populous metropolitan areas, the free-floating bicycle-sharing system (FFBSS) acts as an innovative urban mobility as a service, which provides an ease-of-use feature and extra flexibility in contrast to the traditional shared bicycles with docks. In consideration of customer behaviors, such as abandonment and retrial, which occur in FFBSS, a redistribution strategy for shared bicycles among different user-density locations is presented with an aim to diminish the total operational cost while enhancing the overall service level. To formulate the user and multitype shared bicycle–arrival patterns as nonhomogeneous queues, our results provide a tractable analytical paradigm for a time-varying balancing strategy for FFBSS. The bicycle variation at each virtual zone after each redistribution is determined via a nonstationary queueing model, in which the service time, patience time, and research delay are all subject to general distribution. Then, the bicycle-deployment strategy is evaluated with respect to average queueing length and abandonment rate during a normal workday based on a tailored nonhomogeneous probabilistic matching queue. To verify the efficacy and cost-effectiveness of the proposed bicycle-redistribution strategy, multiple simulation runs are conducted with respect to various times of the day. It shows that the resulting optimal rebalancing strategy is batch-based in synchrony with the time heterogeneity in the traffic demand. Furthermore, several managerial insights are provided to shed light on the rule of thumb in practical FFBSS redistribution coordination.

The generalized join the shortest orbit queue system: stability, exact tail asymptotics and stationary approximations

We introduce the generalized join the shortest queue model with retrials and two infinite capacity orbit queues. Three independent Poisson streams of jobs, namely a smart, and two dedicated streams, flow into a single-server system, which can hold at most one job. Arriving jobs that find the server occupied are routed to the orbits as follows: Blocked jobs from the smart stream are routed to the shortest orbit queue, and in case of a tie, they choose an orbit randomly. Blocked jobs from the dedicated streams are routed directly to their orbits. Orbiting jobs retry to connect with the server at different retrial rates, i.e., heterogeneous orbit queues. Applications of such a system are found in the modeling of wireless cooperative networks. We are interested in the asymptotic behavior of the stationary distribution of this model, provided that the system is stable. More precisely, we investigate the conditions under which the tail asymptotic of the minimum orbit queue length is exactly geometric. Moreover, we apply a heuristic asymptotic approach to obtain approximations of the steady-state joint orbit queue-length distribution. Useful numerical examples are presented and shown that the results obtained through the asymptotic analysis and the heuristic approach are agreed.

Increasing social welfare with delays: strategic customers in the M/G/1 orbit queue

It is well known that strategic customers typically patronize service systems at a higher rate than the socially optimal one. Much literature has focused on inducing customers to patronize such systems at this latter rate. All of this literature considered non-idling policies that are the focus of queueing theory. We demonstrate that strategically imposing delays into such systems can improve the social welfare using the M/G/1 queue with orbit. This queue is a versatile model, that has been extensively studied in the literature from a performance evaluation perspective. In this system, customers who arrive and find the server idle begin immediately their service. However, strategic customers who find the server busy can decide whether to balk or join a virtual queue that is referred to as the orbit. Then, each time the server finishes a service, he begins to retrieve a customer from the orbit and the corresponding retrieving time is not negligible. These retrieving times can be seen as extra delays that are imposed on customers that find a busy server. We further show that when customers are strategic there are certain ranges of the parameters where strategically delaying the orbit customers can increase the welfare of a system or even maximize it. To this end, we characterize and compute the equilibrium strategies for the customers' joining/balking dilemma. We consider both the unobservable and observable versions of the system, compare them and provide some insight on the optimal delay and level of information in such systems.

Equilibrium joining strategies in the single-server constant retrial queues with Bernoulli vacations

We consider the equilibrium joining strategies in an M/M/1 constant retrial queue with Bernoulli vacations. There is no buffer in front of the server, thus an arriving customer will be served immediately if the server is available, and blocked ones wait in a queue if the server is busy or under vacation. The queue length information of orbit is observable to customers upon their arrivals. Then, blocked customers decide whether to join the orbit or not based on a reward-cost structure and their information level. After completing service, the server begins a vacation or remains available and it becomes available again when a vacation ends. The available server seeks to serve the customer in the head of the orbit queue. During the seeking process, an external arrival can interrupt it and obtain service. Our goal is to explore equilibrium behavior of customers in two information cases, fully observable case and almost observable case, which corresponding to whether blocked arrivals can differentiate the state of unavailable server. We obtain the threshold strategies of blocked customers in two information cases and provide numerical experiments to characterize the influence of different parameters on the equilibrium joining strategies.

Analysis of the symmetric join the shortest orbit queue

This work introduces the join the shortest queue policy in the retrial setting. We consider a Markovian single server retrial system with two infinite capacity orbits. An arriving job finding the server busy, it is forwarded to the least loaded orbit. Otherwise, it is forwarded to an orbit randomly. Orbiting jobs of either type retry to access the server independently. We investigate the stability condition, the stationary tail decay rate, and obtain the equilibrium distribution by using the compensation method.

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.

Two-way Communication Orbit Queues with Server Vacation

We consider a single server retrial model with two streams of calls namely, incoming and outgoing calls. Each stream consists of multiple classes of calls. As part of the internal work load, presence of outgoing calls are always assumed in the system. Arrival of incoming calls obey the Poisson law. Upon seeing a busy server at its arrival epoch, an incoming call will be directed to an orbit according to the class it belongs to and tries to get an idle sever in gap of exponential amount of time with class dependent mean. Similar kind of attempt is being made by the outgoing calls also to reach an idle server. Once the sever becomes idle, if neither an incoming nor an outgoing call turns up for an exponential amount of time, the server goes for vacation and the vacation time is assumed to be exponential. Within each stream, service times of multiple classes of calls are assumed to be independent exponential with class dependent means. Matrix Analytic Method and regenerative approach are used to derive the explicit form of the steady state probabilities. Many performance measures are computed to analyse the system performance.

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

In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered. We assume that the customers who find the server busy are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline and only the customer at the head of the queue is allowed access to the server. During the normal period, if the orbit queue is not empty at a service completion instant, the server begins a working vacation with specified probabilityq(0 ≤ q ≤ 1), and with probability 1 − q, he waits for serving the next customer. During the working vacation period, customers can be served at a lower service rate. We first present the necessary and sufficient condition for the system to be stable. Using the supplementary variable method, we deal with the generating functions of the server state and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.

Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking

In this paper, we consider a single-server retrial model with multiple classes of customers. Arrival of customers follow independent Poisson rule. A new customer, facing a busy server upon his arrival, may join the corresponding (class-dependent) orbit queue with a class-dependent probability, or leaves the system forever (balks). The orbit queues follow constant retrial rate discipline, that is, only one (oldest) orbital customer of each orbit queue makes attempts to occupy the server, in a gap of class-dependent exponential times. Within each class, service times are assumed to be independent and identically distributed (iid). We show that this setting generalizes the so-called two-way communication systems.This multiclass system with general service time distributions is analysed using regenerative approach. Necessary and sufficient stability conditions, as well as some explicit expressions for the basic steady-state probabilities, are obtained. A restricted, two-way communication model with exponential service time distributions, is analysed by matrix-analytic method. Moreover, we combine both methods to efficiently derive explicit solutions for the restricted model.An extensive simulation analysis is performed to gain deep insight into the model stability and performance. It is shown that both the simulated and exact results agree on some important measures for which analytical expressions are available, and hence establish the validity of our theoretical treatment. We numerically study the sophisticated dependence structure of the model to uncover the orbits interaction. We give further details and intuitive explanation for the system performance which complements the derived explicit expressions.

On the power series approximations of a structured batch arrival two-class retrial system with weighted fair orbit queues

We provide power series approximations for a structured batch arrival single server retrial system with probabilistic priority and two infinite capacity weighted fair orbit queues, under a queue-based constant retrial policy. We consider both exponential and arbitrary distributed service times. In both cases we obtain power series expansions of the generating functions of the stationary joint orbit queue-length distributions, and provide a recursive approach to calculate their coefficients. Numerical results show the accuracy of our approach, and useful insights in the system performance.

Tail asymptotics for a batch service polling system with retrials and nonpersistent customers

Perel and Yechiali 2014 [16] considered a multi-queue single-server retrial polling system with batch service of an unlimited size, i.e., the so called “Israeli queue” with retrial, where the system consists of a main queue and an orbit queue, and only the customer at the head of the orbit queue is allowed to try to access the main queue. In this present paper, we aim to give a further study on this queueing model, in which each of the retrial customers in the orbit queue independently repeatedly tries for receiving service, and customers may abandon at the arrival instant from outside or at the departure epoch from the orbit queue. By analysing this model, we find that it is difficult to obtain an explicit closed form solution for the joint stationary probability distribution of the number of retrial customers in the orbit queue and the number of groups in the main queue. Therefore, by making use of the matrix analytic approach and the censoring technique, we derive the tail asymptotics result for the stationary joint probabilities.

A TWO-CLASS RETRIAL SYSTEM WITH COUPLED ORBIT QUEUES

We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, typeicustomer,i=1, 2, is routed to a separate typeiorbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queueitries to re-dispatch a blocked customer of typeito the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μito$\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

A queueing model with two classes of retrial customers and paired services

We mathematically investigate a single server system accepting two types of retrial customers and paired service. If upon arrival a customer finds the server busy, it is routed to an infinite capacity orbit queue according to each type. Upon a service completion epoch, if at least one orbit queue is non-empty, the server seeks to find customers from the orbits. If both orbit queues are non-empty, the seeking process will bring to the service area a pair of customers, one from each orbit. If there is only one non-empty, then a single customer from this orbit queue will be brought to the service area. However, if a primary customer arrives during the seeking process it will occupy the server immediately. It is shown that the joint stationary orbit queue length distribution at service completion epochs is determined by solving a Riemann boundary value problem. Stability condition is investigated, while generalizations of the main model are also discussed. A simple numerical example is obtained and yields insight into the behavior of the system.

A repairable retrial queue under Bernoulli schedule and general retrial policy

This paper considers a repairable M/G/1 retrial queue with Bernoulli schedule and a general retrial policy, which is motivated by a contention problem in the downlink direction of wireless base stations in cognitive radio networks. Arriving Customers (called primary arrivals) who cannot receive service upon arrival either join the infinite waiting space in front of the server (called as the normal queue) with probability $$q$$ , or enter the orbit with probability $$1-q$$ according to the FCFS discipline. If the server breaks down in the process of the service of a customer, the customer in service either joins the orbit queue or leaves the system forever. First, we study the ergodicity of two related embedded Markov chains and derive stationary distributions. Second, we find the steady-state joint generating function of the number of customers in both queues. Some important performance measures of the system are obtained. Third, the reliability analysis of the system is also given. Finally, numerical examples are given to illustrate the impact of system parameters on the system performance measures.

Stability analysis of GI/GI/c/K retrial queue with constant retrial rate

We consider a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has \(c\) identical servers and can accommodate up to \(K\) jobs (including \(c\) jobs under service). If a newly arriving job finds the primary queue to be full, it joins the orbit queue. The original primary jobs arrive to the system according to a renewal process. The jobs have i.i.d. service times. The head of line job in the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the length of the orbit queue. Telephone exchange systems, medium access protocols, optical networks with near-zero buffering and TCP short-file transfers are some telecommunication applications of the proposed queueing system. The model is also applicable in logistics. We establish sufficient stability conditions for this system. In addition to the known cases, the proposed model covers a number of new particular cases with the closed-form stability conditions. The stability conditions that we obtained have clear probabilistic interpretation.

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 $$M$$ 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.