Peak congestion in multi-server service systems with slowly varying arrival rates

In this paper we consider the M _t queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for M _t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak. This revised version was published online in June 2006 with corrections to the Cover Date.     

Queues with impolite customers

An interesting behavior of customers arriving to a queue for service concerns the manner in which they join the queue. The arrival discipline of the customers may be impolite, in the sense that an arriving customer who finds all servers busy may pick a position which is not necessarily at the end of the line. We introduce and discuss in detail such an arrival discipline of sufficient generality which has interesting applications. In particular, we show that the more impolite an arrival discipline is, the bigger is the variance of the waiting time. We also study a special model in more depth to provide simple computational formulas for several performance measures.     

Waiting-Time Distributions in Polling Systems with Simultaneous Batch Arrivals

We study the delay in asymmetric cyclic polling models with general mixtures of gated and exhaustive service, with generally distributed service times and switch-over times, and in which batches of customers may arrive simultaneously at the different queues. We show that (1–)X _i converges to a gamma distribution with known parameters as the offered load tends to unity, where X _i is the steady-state length of queue i at an arbitrary polling instant at that queue. The result is shown to lead to closed-form expressions for the Laplace–Stieltjes transform (LST) of the waiting-time distributions at each of the queues (under proper scalings), in a general parameter setting. The results show explicitly how the distribution of the delay depends on the system parameters, and in particular, on the simultaneity of the arrivals. The results also suggest simple and fast approximations for the tail probabilities and the moments of the delay in stable polling systems, explicitly capturing the impact of the correlation structure in the arrival processes. Numerical experiments indicate that the approximations are accurate for medium and heavily loaded systems.     

An analytical solution for the discrete time single server system with semi-Markovian arrivals

In this paper we derive an analytical solution for the stationary distribution of the number of customers and the idle time in a single server system with semi-Markovian arrival processes in discrete time domain (SM/G/1). This kind of arrival process enables us to take autocorrelations into account, with various applications for the modeling of communication and manufacturing systems. It will be shown that the distribution of the customer number can be represented as a linear combination of geometric distributions. Thus a simple calculation of higher moments of the customer number is possible.     

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.     

Class dependent departure process from multiclass phase queues: Exact and approximate analyses

In this paper we study class dependent departure processes from phase type queues. When the arrival process for a subset of the classes is a Poisson process, we determine the Laplace-Stieltjes transform of the stationary inter-departure times of the combined output of all the other classes. We also propose and test approximations for the squared coefficient of variation of the stationary inter-departure times of each customer class. The approximations are based on the detailed structure of the second order measures of the aggregate departure process. Finally, we propose renewal approximations for the class dependent departure process that take into account the utilization of the queue that customers next visit.     

A Markov-modulated M/G/1 queue I: Stationary distribution

We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Optimal admission policies for a finite queue with bursty arrivals

We consider a service system with a single server, a finite waiting room and two classes of customers with deterministic service time. Primary jobs arrive at random and are admitted whenever there is room in the system. At the beginning of each period, secondary jobs can be admitted from an infinite pool. A revenue is earned upon admission of each job, with the primary jobs bringing a higher contribution than the secondary jobs, the objective being to maximize the total discounted revenue over an infinite horizon. We model the system as a discrete time Markov decision process and show that a monotone admission policy is optimal when the number of primary arrivals has a fixed distribution. Moreover, when the primary arrival distribution varies with time according to a finite state Markov chain, we show that the optimal policy is conditionally monotone and that the numerical computation of an optimal policy, in this case, is substantially more difficult than in the case of stationary arrivals.This research was supported in part by the National Science Foundation, under grant ECS-8803061, while the author was at the University of Arizona.     

Markov-modulatedPH/G/1 queueing systems

We study a PH/G/1 queue in which the arrival process and the service times depend on the state of an underlying Markov chain J(t) on a countable state spaceE. We derive the busy period process, waiting time and idle time of this queueing system. We also study the Markov modulated E_K/G/1 queueing system as a special case.     

Optimal strategy policy in batch arrival queue with server breakdowns and multiple vacations

This paper considers a like-queue production system in which server vacations and breakdowns are possible. The decision-maker can turn a single server on at any arrival epoch or off at any service completion. We model the system by an M^[x]/M/1 queueing system with N policy. The server can be turned off and takes a vacation with exponential random length whenever the system is empty. If the number of units waiting in the system at any vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N units in the system, he immediately starts to serve the waiting units. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. We derive the distribution of the system size through the probability generating function. We further study the steady-state behavior of the system size distribution at random (stationary) point of time as well as the queue size distribution at departure point of time. Other system characteristics are obtained by means of the grand process and the renewal process. Finally, the expected cost per unit time is considered to determine the optimal operating policy at a minimum cost. The sensitivity analysis is also presented through numerical experiments.     

Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.     

Control and recovery from rare congestion events in a large multi-server system

We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multi-server system in which customer holding times have a general distribution. There are two cases, depending on whether or not we exploit the age distribution (the distribution of elapsed holding times of customers in service). If we do not exploit the age distribution, then the rare congestion event is a large number of customers present. If we do exploit the age distribution, then the rare event is an unusual age distribution, possibly accompanied by a large number of customers present. As an approximation, we represent the large multi-server system as an M/G/∞ model. We prove that, under regularity conditions, the fluid approximations are asymptotically correct as the arrival rate increases. The fluid approximations show the impact upon the recovery time of the holding-time distribution beyond its mean. The recovery time may or not be affected by the holding-time distribution having a long tail, depending on the precise definition of recovery. The fluid approximations can be used to analyze various overload control schemes, such as reducing the arrival rate or interrupting services in progress. We also establish large deviations principles to show that the two kinds of rare events have the same exponentially small order. We give numerical examples showing the effect of the holding-time distribution and the age distribution, focusing especially on the consequences of long-tail distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Shortest Queue Version of the Erlang Loss Model

We consider two parallel M / M / N / N queues. Thus there are N servers in each queue and no waiting line(s). The network is fed by a single Poisson arrival stream of rate λ, and the 2 N servers are identical exponential servers working at rate μ. A new arrival is routed to the queue with the smaller number of occupied servers. If both have the same occupancy then the arrival is routed randomly, with the probability of joining either queue being 1/2. This model may be viewed as the shortest queue version of the classic Erlang loss model. If all 2 N servers are occupied further arrivals are turned away and lost. We let  ρ=λ/μ  and   a = N /ρ= N μ/λ  . We study this model both numerically and asymptotically. For the latter we consider heavily loaded systems (ρ→∞) with a comparably large number of servers (   N →∞  with   a = O (1))  . We obtain asymptotic approximations to the joint steady state distribution of finding m servers occupied in the first queue and n in the second. We also consider the marginal distribution of the number of occupied servers in the second queue, as well as some conditional distributions. We show that aspects of the solution are much different according as   a > 1/2, a ≈ 1/2, 1/4 < a < 1/2, a ≈ 1/4  or  0 < a < 1/4  . The asymptotic approximations are shown to be quite accurate numerically.     

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

An explicit solution to a tandem queueing model

We consider two queues in tandem, each with an exponential server, and with deterministic arrivals to the first queue. We obtain an explicit solution for the steady state distribution of the process (N₁(t), N₂(t), Y(t)), where N_j(t) is the queue length in the jth queue and Y(t) measures the time elapsed since the last arrival. Then we obtain the marginal distributions of (N₁(t), N₂(t)) and of N₂(t). We also evaluate the solution in various limiting cases, such as heavy traffic. This revised version was published online in June 2006 with corrections to the Cover Date.     

A comparison of the sliding window and the leaky bucket

In this paper we compare the sliding-window (SW) and leaky-bucket (LB) input regulators. These regulators reject, or treat as lower priority, certain arrivals to a queueing system, so as to reduce congestion in the queueing system. Such regulators are currently of interest for access control in emerging high-speed communication networks. The SW admits no more than a specified numberW of arrivals in any interval of specified lengthL. The LB is a counter that increases by one up to a maximum capacityC for each arrival and decreases continuously at a given drain rate to as low as zero; an arrival is admitted if the counter is less than or equal toC–1. To indirectly represent the impact of the regulator on the performance of the queueing system, we focus on the maximum bursts admissible at the peak rate. We show that the SW admits larger bursts than the LB at any given peak rate and admissible average rate. To make the comparison, we use a special construction: We start with a sample path of an arrival process with a given peak rate. We choose a window lengthL for the SW and find the minimum window contentW that is just conforming (so there are no rejections). We then set the LB drain rate equal toW/L, so that the two admissible average rates are identical. Finally, we choose the LB capacityC so that the given arrival process is also just conforming for the LB. With this construction, we show that the SW will admit larger bursts at the peak rate than the LB. We also develop approximations for these maximum burst sizes and their ratio over long time intervals based on extreme-value asymptotics. We use simulations to confirm that these approximations do indeed enable us to predict the burst ratios for typical stochastic arrival processes.     

Integer Valued Autoegressive Process with Katz Arrivals and Its Application in Predicting the Count of the Cases of Respiratory Disease

The traditional PAR process (Poisson autoregressive process) assumes that the arrival process is the equi-dispersed Poisson process, with its mean being equal to its variance. Whereas the arrival process in the real DGP (data generating process) could either be over-dispersed, with variance being greater than the mean, or under-dispersed, with variance being less than the mean. This paper proposes using the Katz family distributions to model the arrival process in the INAR process (integer valued autoregressive process with Katz arrivals) and deploying Monte Carlo simulations to examine the performance of maximum likelihood (ML) and method of moments (MM) estimators of INAR-Katz model. Finally, we used the INAR-Katz process to model count data of hospital emergency room visits for respiratory disease. The results show that the INAR-Katz model outperforms the Poisson model, PAR(1) model, and has great potential in empirical application.     

Robust berth scheduling with uncertain vessel delay and handling time

In container terminals, the actual arrival time and handling time of a vessel often deviate from the scheduled ones. Being the input to yard space allocation and crane planning, berth allocation is one of the most important activities in container terminals. Any change of berth plan may lead to significant changes of other operations, deteriorating the reliability and efficiency of terminal operations. In this paper, we study a robust berth allocation problem (RBAP) which explicitly considers the uncertainty of vessel arrival delay and handling time. Time buffers are inserted between the vessels occupying the same berthing location to give room for uncertain delays. Using total departure delay of vessels as the service measure and the length of buffer time as the robustness measure, we formulate RBAP to balance the service level and plan robustness. Based on the properties of the optimal solution, we develop a robust berth scheduling algorithm (RBSA) that integrates simulated annealing and branch-and-bound algorithm. To evaluate our model and algorithm design, we conduct computational study to show the effectiveness of the proposed RBSA algorithm, and use simulation to validate the robustness and service level of the RBAP formulation.     

A note on some relations in the queue Glx/M/c

Closed-form relations are derived for the probabilities and performance measures observed at random/arrival/departure epochs in a multi-server queue with group arrivals.