Two-server Queueing Research Articles

A Two-Server Queue with Interdependence between Arrival and Service Processes

In this paper, we analyse a queueing system with two servers where the arrival and service processes are interdependent. The evolution of these processes is governed by transitions on the product space of three Markov chains, which are descriptors of the arrival and service processes. The transitions in this Markov chain follow a semi-Markov rule and the exponential distribution governs the sojourn times in the states. The stability condition of the system is derived and the stationary distribution is calculated for the system in equilibrium. Several important performance measures are provided, and numerical illustrations of the model are presented.

The impact of line-sitting on a two-server queueing system

In the presence of multiple lines, customers can often occupy the lines simultaneously with the help of someone else, so as to reduce their wait times. To understand the impact of such line-sitting behavior on the service system with multiple lines, we establish a queueing game-theoretic model to capture the interactions among customers, line-sitting firm and service provider. In particular, we consider a congested-prone service system with two dedicated queues, where arriving customers can join the two queues simultaneously by hiring a line-sitter to wait in one of the line on behalf of them, enabling them to receive service in any one of the queues. Our main results are as follows. First, contrary to our conventional wisdom that line-sitting firm could be more profitable when the system is more congested, we find instead that a queue with an intermediate demand volume can be most susceptible to line-sitting, whereas queues with either a very small or a very large demand volume may not be financially attractive to line-sitters. Second, our result discloses that the provision of line-sitting service in multiple lines can efficiently increase the system throughput. In the short run, both revenue of service provider and the customer surplus can be improved with the provision of line-sitting, which leads to a win-win situation. However, in the long run, line-sitting will stimulate a higher service fee charged by the service provider, reversing its short-run beneficial effect on customers, which results in a lower customer surplus. To restore the efficiency of line-sitting, we propose a centralized regulation policy from the perspective of social planner to maximize the social welfare without detriment to the customer population.

Optimal Policy for Controlling Two-Server Queueing Systems with Jockeying

This paper studies the optimal policy for joint control of admission, routing, service, and jockeying in a queueing system consisting of two exponential servers in parallel. Jobs arrive according to a Poisson process. Upon each arrival, an admission/routing decision is made, and the accepted job is routed to one of the two servers with each being associated with a queue. After each service completion, the servers have an option of serving a job from its own queue, serving a jockeying job from another queue, or staying idle. The system performance is inclusive of the revenues from accepted jobs, the costs of holding jobs in queues, the service costs and the job jockeying costs. To maximize the total expected discounted return, we formulate a Markov decision process (MDP) model for this system. The value iteration method is employed to characterize the optimal policy as a hedging point policy. Numerical studies verify the structure of the hedging point policy which is convenient for implementing control actions in practice.

Recurrence of two-dimensional queueing processes, and random walk exit times from the quadrant

Let X=(X1,X2) be a two-dimensional random variable and X(n), n∈N, a sequence of i.i.d. copies of X. The associated random walk is S(n)=X(1)+⋯+X(n). The corresponding absorbed-reflected walk W(n), n∈N, in the first quadrant is given by W(0)=x∈R+2 and W(n)=max{0,W(n−1)−X(n)}, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on X. It turns out that this is directly related with the tail asymptotics of the exit time of the random walk x+S(n) from the quadrant, so that the main part of this paper is devoted to an analysis of that exit time in relation with the drift vector, that is, the expectation of X.

A queueing inventory system with threshold based priority service and a flexible server

In this paper, we analyze two-server queueing - inventory system which consists of two-parallel queues and two category of customers i.e., high and low priority customers. The high priority customer generates demand for an item whereas the low priority customers arrive only for repair (not demanding items). Server 1 offers his service to first queue which involves only high priority customers and server 2 is a flexible server who offers his service to both the queues. When there is no availability of service in both the queues, server 1 will be at idle state whereas server 2 goes on vacation. We find a priority service rule to minimize the loss probability of low priority customers. The low priority customers will be served on threshold-based service such that a threshold L which is placed in the second queue and the service starts only if the number customers in the second queue exceeds the threshold level. The limiting probability distribution of the four random variables is computed in steady state. Also, some significant change in measures of the system are derived. Finally, the suitable cost function is defined and significant numerical examples are projected.

Modelling and optimisation of a two-server queue with multiple vacations and working breakdowns

This paper presents a steady-state analysis of an M/M/2 queue with heterogeneous servers (Server 1 and Server 2). Server 1 is reliable and may leave for a vacation when the system becomes empty. Sever 2 is unreliable and may break down while serving customers. When a breakdown occurs, Server 2 reduces the service rate rather than halting service. We formulate this queueing model as a quasi birth-and-death (QBD) process, using the matrix geometric method to compute the stationary distribution of system size. We also develop several measures to evaluate the performance of the system. A cost model based on system performance measures is formulated as a heuristic cost optimisation problem subject to stability conditions. A canonical particle swarm optimisation algorithm is used to obtain numerical solutions for the approximate optimal service rates of Server 1 and Server 2. Moreover, we present numerical results showing the effects of various parameters on the approximate optimal service rates as well as a practical example illustrating the application of the proposed model.

A Two-Server Queue with Mutual Consultations

This paper analyses a two-server queueing model with consultation between the servers. Each of the servers provides consultation to the other server with a preemptive priority over customers. After consultation, the services in progress will be resumed. The arrivals to the system follows a Poisson process and requirement of consultations follow mutually independent Poisson processes; the duration of consultations follow mutually independent and identical exponential distributions. The service times of the customers at the servers are assumed to follow mutually independent phase type distributions. An explicit expression for the system stability is obtained. System performance indices are computed and numerical illustrations are provided. Further we consider two particular cases of the model described. Finally, we compare these through numerical investigations.

Specialized and flexible servers subject to the effects of learning and forgetting

We consider learning and forgetting in the context of two-server queueing systems and evaluate the tradeoff between utilizing specialized and flexible servers. Specifically, we investigate the performance of two queueing systems. The first system utilizes a specialized workforce where every server handles one job type. The specialized workforce splits the system into two queues where the dedicated servers capitalize on the learning process and consequently reduce the service time. The second system utilizes a flexible workforce where a server can handle any job type. The flexible workforce system allows for the servers to be arranged in parallel where alternating job types results in forgetting and accordingly an increase in service time. A numerical study investigates the impact of the workforce policies on the system performance measures.

Cascade equilibrium strategies in a two-server queueing system with inspection cost

Time-sensitive customers arriving to a system of parallel servers will search for the least congested queue. However, customers do not always have full information of the system state, because the search is associated with a cost. We present a model of parallel queues in front of two servers that provide the same service. Upon arrival, each customer inspects the queue in front of one server before either joining it or inspecting the other. If both queues were inspected, the customer joins the inspected queue that minimizes sojourn time. The solution of this model is not straightforward, even when the system contains only two servers, and the equilibrium is not always a threshold strategy. We show that, in many cases, a unique equilibrium strategy that contains cascades exists: customers choose one action (join or inspect) when they observe i and i+2 customers in the first observed queue, and the other action when they observe i+1 customers in the first observed queue. We find cascade equilibrium strategies even when the servers are identical with respect to service rate or inspection cost. We also show that compared to the case where all customers first inspect the same queue, symmetric inspection reduces system load but not necessarily customers’ expected cost.

Equilibrium queueing strategies of two types of customers in a two-server queue

We consider a single queue with two identical servers and two types of customers. The high-type customer is more delay-sensitive but brings less workload to the system than the low-type customer. We obtain the equilibrium queueing strategy for each type of customers.

Unreliable two-server queueing system with variable repair rates

This article considers an infinite-capacity two-server queue system, where the servers are subject to failure at any time. A repairman is responsible for repair the failure server and the repair rate is changeable depending on the number of failure servers. The repair rate of only one server is failure and that of two servers are all failure may be different. The steady-state reliability indexes and the stability condition of queue are derived through the transfer rate matrix. The expected queue length and some related steady-state performances are obtained by the probability-generating function transform method. Numerical examples are provided for illustration purpose.

Heterogeneous two-server queueing system with reverse balking and reneging

This paper presents an analysis of heterogeneous two-server queueing system with feedback, reverse balking, reneging and retention of reneged customers. The steady-state solution of the model is obtained and different measures of effectiveness are derived. Economic analysis of the model is performed studying the impact of balking probability and retention probability on the total expected profit of the system.

Routing in a queueing system with two heterogeneous servers in speed and in quality of resolution

ABSTRACTHeterogeneous servers, in manufacturing and service systems, may have different speeds and different quality levels for the provided service or good For a two-server queueing model, we formulate the job routing problem for minimizing the stationary weighted sum of the expected time spent in the system and the number of unsatisfied customers per time unit. Using a Markov decision process approach, we prove that the optimal routing policy of jobs to service is a threshold policy that depends on the queue length. When the number of waiting jobs in the queue is below a certain threshold, only one server should work and the other one remains idle. At or above this threshold, both servers should serve jobs. This is an extension of the known result where only the heterogeneity in speed is considered.

Stability of a queue with discriminatory random order service discipline and heterogeneous servers

We consider a queueing system with two classes of customers, two heterogeneous servers, and discriminatory random order service (DROS) discipline. The two servers may have either the same or different DROS weights for each class. Customers of each class arrive according to a Poisson process and the service times of each class of customers are assumed to be exponentially distributed with service rate depending on both the customer's class and the servers. We provide stability and instability conditions for this two-class two-server queue with DROS discipline.

Consolidating or non-consolidating queues: A game theoretic queueing model with holding costs

We consider a two-server queueing system in which the servers choose their service rate based on the demand and holding cost allocation scheme offered by the demand generating entity. We provide an optimal holding cost allocation scheme that leads to the maximum possible service rate for each of a pooled and a split system. Our results suggest that careful allocation of holding costs can create incentives that enable minimum turnaround times using a common queue.

A Queueing System with Two Parallel Lines, Cost-Conscious Customers, and Jockeying

We consider a two-server queueing system with the special feature of cost-conscious and jockeying customers. It is assumed that a new arriving customer joins one of the two queues only if the associated costs for him do not exceed a given upper threshold c. These costs are given by a linear combination of the current queue length at the chosen server and a service fee. Thus, if N i are the current queue lengths, β i > 0 are the service fees and α i > 0 are certain prespecified weights for the two servers (i ∈ {1, 2}), the customer enters the system if and only if min [α1 N 1 + β1, α2 N 2 + β2] ≤ c, and in this case he goes to server 1 or 2 depending on which of the two terms is smaller; if equality occurs, he selects each server with probability 1/2. A customer waiting in line jockeys if, due to a departure from the other line during his waiting time, his residual costs become larger than the costs he will have after switching to the other server. In this model, new customers do not necessarily join the shortest queue (if they enter the system at all) and customers may jockey to a longer queue (if such a move reduces residual costs). Under the assumption that the arrival times form a Poisson process and the service times are exponential we derive the stationary distribution of the two-dimensional Markov process of queue lengths.

Waiting time distribution in a two-class two-server heterogeneous priority queue

Our aim is to analyze a Markovian two-class two-server queue with non-preemptive heterogeneous priority structures. We use matrix-geometric methods and derive the stationary distribution of queue lengths. To obtain the distribution of waiting times, we develop a dedicated algorithm by extending the approach of Ramaswami and Lucantoni.

Sensitivity formulas of performance in two-server cyclic queueing networks with phase-type distributed service times

With the approach of the infinitesimal generator perturbation, sensitivities of the steady-state performance are studied in two-server cyclic queueing networks with phase-type distributed service times. Sensitivity formulas expressed by the potentials of the networks are given for parameter-dependent performance functions. An algorithm to compute the potentials and the performance derivatives of the networks is given, and a numerical example is provided.

A simple proof of the optimality of a threshold policy in a two-server queueing system

Lin and Kumar (1984) introduced a control model with a single queue and two heterogeneous servers. They showed, using policy iteration, that the slower server should only be used if the queue length is above a certain level, i.e., the optimal policy is of threshold type. In this note we give a simple iterative proof of this result.

Expected number analysis of a two-server queueing network subject to inter-stage inspection and rework

In this paper, an attempt has been made to incorporate the concept of rework in the probabilistic analysis of queueing systems. The paper provides an integrated framework to consider servicing, inspection and rework activities simultaneously in two-stage queueing systems. The purpose of this paper is to analyse some of the transient-state characteristics of a two-server queueing system subject to an initial buffer of finite capacity, inter-stage inspection and rework. A stochastic model is developed to analyse the system and explicit analytical expressions for some of the system characteristics have been obtained using the statespace method and regeneration point technique. All the random variables involved in the analysis are assumed to be arbitrary except for the arrival time distribution to the initial buffer which is assumed to follow exponential distribution.