In this paper we examine three different approximation techniques for modeling packet loss in finite-buffer voice multiplexers. The performance models studied differ primarily in the manner in which the superposition of the voice sources (i.e., the arrival process) is modeled. The first approach models the superimposed voice sources as a renewal process and performance calculations are based only on the first two moments of the renewal process. The second approach is based on modeling the superimposed voice sources as a Markov Modulated Poisson Process (MMPP). Our choice of parameters for the MMPP attempts to capture aspects of the arrival process in an alternate, more intuitive, manner than previously proposed approaches for determining the MMPP parameters and is shown to compute loss more accurately. Finally, we also evaluate a fluid flow approximation for computing packet loss. For all three approaches, we consider as a unifying example, the case of multiplexing voice sou...

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: We study the long-run “state” behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both “actual ” and “virtual ” delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural “explanations ” for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Beneˇs’s formula for the delay distribution in M/G/1. We also discuss how to extend our results to settings more general than GI/G/1.

We consider the problem of minimizing staffing costs in an inbound call center, while main-taining an acceptable level of service in multiple time periods. The problem is complicated by the fact that staffing level in one time period can affect the service levels in subsequent periods. Moreover, staff schedules typically take the form of shifts covering several periods. Interactions between staffing levels in different time periods, as well as the impact of shift requirements on the staffing levels and cost should be considered in the planning. Traditional staffing methods based on stationary queueing formulas do not take this into account. We present a simulation-based analytic center cutting plane method to solve a sample average approximation of the problem. We establish convergence of the method when the service level functions are discrete pseudoconcave. An extensive numerical study of a moderately large call center shows that the method is robust and, in most of the test cases, outperforms traditional staffing heuristics that are based on analytical queueing methods.

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic limiting regimes.

We consider a system of N queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues); and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zeroswitchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzeroswitchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [8] and Cooper, Niu, and Srinivasan [4]. Key words. Polling models, cyclic queues, waiting times, decomposition, switchover times, vacation models. 1

timescales QoS criteria Application to the network

Abstract not found

We use computer simulation to study the performance of alternative real-time delay estimators in heavily loaded multiserver queueing models. These delay estimates may be used to make delay announcements in call centers and related service systems. We consider the classical delay estimator based on the queue length, QLs, which multiplies the queue length plus one times the mean interval between successive service completions, ignoring customer abandonment. We show that QLs has a superior performance in the GI/M/s model, but that there is a need to go beyond it in the GI/GI/s + GI model, allowing abandonment. To this end, we propose new, simple and effective, delay estimators based on the queue length. We also consider a delay estimator based on recent customer delay history in the system: the delay of the last customer to enter service, LES. 1

Higher-order distributional properties in closed queueing networks

We consider curbside bus stops of the kind that serve multiple bus routes, and that are isolated from the effects of traffic signals and other stops. A Markov chain embedded in the bus queueing process is used to develop steady-state queueing models for two special cases of this stop type. The models estimate the maximum rate that buses can arrive to, and be served by the stop, and still satisfy a specified target of average bus delay. These models can be used to determine, for example, a stop’s suitable number of bus berths, given the bus demand and the specified delay target. The solutions for the two special cases are used to derive a closed-form, parsimonious approximation model for general cases. Our approximations closely match simulations for a range of conditions that arise in real settings. And the approximations unveil how suitable choices for the number of bus berths are influenced by both, the variation in the time that buses spend serving passengers at the stop, and the specified delay target. The models further show why the proxy measure that is used for the delay target in other bus-stop studies is a poor one.