First-come First-served Service Discipline Research Articles

$$G/{ GI}/N(+{ GI})$$ queues with service interruptions in the Halfin–Whitt regime

We study \(G/GI/N(+GI)\) queues with alternating renewal service interruptions in the Halfin–Whitt regime. The systems experience up and down alternating periods. In the up periods, the systems operate normally as the usual \(G/GI/N(+GI)\) queues with non-idling first-come–first-served service discipline. In the down periods, arrivals continue entering the systems, but all servers stop functioning while the amount of service that each customer has received will be conserved and services will resume when the next up period starts. For models with abandonment, interruptions do not affect customers’ patience times. We assume that the up periods are of the same order as the service times but the down periods are asymptotically negligible compared with the service times. We establish the functional central limit theorems for the queue-length processes and the virtual-waiting time processes in these models, where the limit processes are represented as stochastic integral convolution equations driven by jump processes. The convergence in these limit theorems is proved in the space \({\mathbb D}\) endowed with the Skorohod \(M_1\) topology.

Efficiency-Driven Heavy-Traffic Approximations for Many-Server Queues with Abandonments

To provide useful practical insight into the performance of service-oriented (non-revenue-generating) call centers, which often provide low-to-moderate quality of service, this paper investigates the efficiency-driven (ED), many-server heavy-traffic limiting regime for queues with abandonments. Attention is focused on the M/M/s/r + M model, having a Poisson arrival process, exponential service times, s servers, r extra waiting spaces, exponential abandon times (the final +M), and the first-come–first-served service discipline. Both the number of servers and the arrival rate are allowed to increase, while the individual service and abandonment rates are held fixed. The key is how the two limits are related: In the now common quality-and-efficiency-driven (QED) or Halfin-Whitt limiting regime, the probability of initially being delayed approaches a limit strictly between 0 and 1, while the probability of eventually being served (not abandoning) approaches 1. In contrast, in the ED limiting regime, the probability of eventually being served approaches a limit strictly between 0 and 1, while the probability of initially being delayed approaches 1. To obtain the ED regime, it suffices to let the arrival rate and the number of servers increase with the traffic intensity ρ held fixed with ρ > 1 (so that the arrival rate exceeds the maximum possible service rate). The ED regime can be realistic because with the abandonments, the delays need not be extraordinarily large. When the ED appropriations are appropriate, they are appealing because they are remarkably simple.

Output analysis of a single-buffer multiclass queue: FCFS service

We consider an infinite capacity buffer where the incoming fluid traffic belongs to K different types modulated by K independent Markovian on-off processes. The kth input process is described by three parameters: (λk, μk, rk), where 1/λk is the mean off time, 1/μk is the mean on time, and rk is the constant peak rate during the on time. The buffer empties the fluid at rate c according to a first come first served (FCFS) discipline. The output process of type k fluid is neither Markovian, nor on-off. We approximate it by an on-off process by defining the process to be off if no fluid of type k is leaving the buffer, and on otherwise. We compute the mean on time τkon and mean off time τkoff. We approximate the peak output rate by a constant rko so as to conserve the fluid. We approximate the output process by the three parameters (λko, μko, rko), where λko = 1/τkoff, and μko = 1/τkon. In this paper we derive methods of computing τkon, τkoff and rko for k = 1, 2,…, K. They are based on the results for the computation of expected reward in a fluid queueing system during a first passage time. We illustrate the methodology by a numerical example. In the last section we conduct a similar output analysis for a standard M/G/1 queue with K types of customers arriving according to independent Poisson processes and requiring independent generally distributed service times, and following a FCFS service discipline. For this case we are able to get analytical results.

Output analysis of a single-buffer multiclass queue: FCFS service

We consider an infinite capacity buffer where the incoming fluid traffic belongs to K different types modulated by K independent Markovian on-off processes. The kth input process is described by three parameters: (λ k , μ k , r k ), where 1/λ k is the mean off time, 1/μ k is the mean on time, and r k is the constant peak rate during the on time. The buffer empties the fluid at rate c according to a first come first served (FCFS) discipline. The output process of type k fluid is neither Markovian, nor on-off. We approximate it by an on-off process by defining the process to be off if no fluid of type k is leaving the buffer, and on otherwise. We compute the mean on time τ k on and mean off time τ k off. We approximate the peak output rate by a constant r k o so as to conserve the fluid. We approximate the output process by the three parameters (λ k o, μ k o, r k o), where λ k o = 1/τ k off, and μ k o = 1/τ k on. In this paper we derive methods of computing τ k on, τ k off and r k o for k = 1, 2,…, K. They are based on the results for the computation of expected reward in a fluid queueing system during a first passage time. We illustrate the methodology by a numerical example. In the last section we conduct a similar output analysis for a standard M/G/1 queue with K types of customers arriving according to independent Poisson processes and requiring independent generally distributed service times, and following a FCFS service discipline. For this case we are able to get analytical results.

An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Levy process. When the Levy process representing the net input has no negative jumps, the steady-state distribution of the reflected Levy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated.

Predicting Queueing Delays

This paper investigates the possibility of predicting each customer's waiting time in queue before starting service in a multiserver service system with the first-come first-served service discipline, such as a telephone call center. A predicted waiting-time distribution or an appropriate summary statistic such as the mean or the 90th percentile may be communicated to the customer upon arrival and possibly thereafter in order to improve customer satisfaction. The predicted waiting-time distribution may also be used by the service provider to better manage the service system, e.g., to help decide when to add additional service agents. The possibility of making reliable predictions is enhanced by exploiting information about system state, including the number of customers in the system ahead of the current customer. Additional information beyond the number of customers in the system may be obtained by classifying customers and the service agents to which they are assigned. For nonexponential service times, the elapsed service times of customers in service can often be used to advantage to compute conditional-remaining-service-time distributions. Approximations are proposed to convert the distributions of remaining service times into the distribution of the desired customer waiting time. The analysis reveals the advantage from exploiting additional information.

A conservation law for single-server queues and its applications

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

A conservation law for single-server queues and its applications

We establish a conservation law forG/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in aG/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in aGI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

Path delays in communication networks

A communication network is modelled by a weighted graph. The vertices of the graph represent stations with storage capabilities, while the edges of the graph represent communication channels (or other information processing media). Channel capacity weights are assigned to the edges of the network. The network is assumed to operate in a store-and-forward manner, so that when a channel is busy the messages directed into it are stored at the station, joining there a queue which is governed by a first-come first-served service discipline.

Communication networks: Message path delays

A communication network is modeled by a weighted graph. The vertices of the graph represent stations with storage capabilities, while the edges of the graph represent communication channels (or other information processing media). Channel capacity weights are assigned to the edges of the network. The network is assumed to operate in a store-and-forward manner, so that when a channel is busy the messages directed into it are stored at the station, where it joins a queue that is governed by a first-come first-served service discipline. Assuming that fixed-length messages arrive at random at the network, following the statistics of a Poisson point process, we solve for the steady-state distributions of the message overall delay time, for the average message waiting times at the individual stations, for the average memory size requirements at the stations, as well as for other statistical characteristics of the message flow along a communication path.