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44
Staffing of timevarying queues to achieve timestable performance
, 2005
"... Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probab ..."
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Cited by 28 (19 self)
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Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probability of delay. That procedure tends to produce timestable performance for several other operational measures. Motivated by telephone call centers, we focus on manyserver models with customer abandonment, especially the Markovian Mt/M/st + M model, having an exponential timetoabandon distribution (the +M), an exponential servicetime distribution and a nonhomogeneous Poisson arrival process. We develop three different methods for staffing, with decreasing generality and decreasing complexity: First, we develop a simulationbased iterativestaffing algorithm (ISA), and conduct experiments showing that it is effective. The ISA is appealing because it applies to very general models and is automatically validating: we directly see how well it works. Second, we extend the squarerootstaffing rule, proposed by Jennings et al., which is based on the associated infiniteserver model. The rule dictates that the staff level at time t be st = mt + β √ mt, where mt is the offered load (mean number of busy servers in the infiniteserver model) and the constant β reflects the service grade. We show that the service grade β in the staffing formula can be represented as a function of the target delay probability α by
Simulation run lengths to estimate blocking probabilities
 ACM Transactions on Modelling and Computer Simulation
, 1996
"... We derive formulas approximating the asymptotic variance of four estimators for the steadystate blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision befor ..."
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Cited by 24 (19 self)
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We derive formulas approximating the asymptotic variance of four estimators for the steadystate blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision before the simulation has been run, which can aid in the design of simulation experiments. They also indicate that one estimator can be much better than another, depending on the loading. An indirect estimator based on estimating the mean occupancy is significantly more (less) efficient than a direct estimator for heavy (light) loads. A major concern is the way computational effort scales with system size. For all the estimators, the asymptotic variance tends to be inversely proportional to the system size, so that the computational effort (regarded as proportional to the product of the asymptotic variance and the arrival rate) does not grow as system size increases. Indeed, holding the blocking probability fixed, the computational effort with a good estimator decreases to 0 as the system size increases. The asymptotic variance formulas also reveal the impact of the arrivalprocess and servicetime variability on the statistical precision. We validate these formulas by comparing them to exact numerical
Estimating the parameters of a nonhomogeneous Poisson process with linear rate
 Telecommunication Systems
, 1996
"... Motivated by telecommunication applications, we investigate ways to estimate the parameters of a nonhomogeneous Poisson process with linear rate over a finite interval, based on the number of counts in measurement subintervals. Such a linear arrivalrate function can serve as a component of a piecew ..."
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Cited by 24 (13 self)
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Motivated by telecommunication applications, we investigate ways to estimate the parameters of a nonhomogeneous Poisson process with linear rate over a finite interval, based on the number of counts in measurement subintervals. Such a linear arrivalrate function can serve as a component of a piecewiselinear approximation to a general arrivalrate function. We consider ordinary least squares (OLS), iterative weighted least squares (IWLS) and maximum likelihood (ML), all constrained to yield a nonnegative rate function. We prove that ML coincides with IWLS. As a reference point, we also consider the theoretically optimal weighted least squares (TWLS), which is least squares with weights inversely proportional to the variances (which would not be known with data). Overall, ML performs almost as well as TWLS. We describe computer simulations conducted to evaluate these estimation procedures. None of the procedures differ greatly when the rate function is not near 0 at either end, but when the rate function is near 0 at one end, TWLS and ML are significantly more effective than OLS. The number of measurement subintervals (with fixed total interval) makes surprisingly little difference when the rate function is not near 0 at either end. The variances are higher with only two or three
Sensitivity to the servicetime distribution in the nonstationary Erlang loss model
 Management Sci
, 1995
"... The stationary Erlang loss model is a classic example of an insensitive queueing system: The steadystate distribution of the number of busy servers depends on the servicetime distribution only through its mean. However, when the arrival process is a nonstationary Poisson process, the insensitivity ..."
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Cited by 19 (10 self)
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The stationary Erlang loss model is a classic example of an insensitive queueing system: The steadystate distribution of the number of busy servers depends on the servicetime distribution only through its mean. However, when the arrival process is a nonstationary Poisson process, the insensitivity property is lost. We develop a simple effective numerical algorithm for the M t/PH/s/0 model with two service phases and a nonhomogeneous Poisson arrival process, and apply it to show that the timedependent blocking probability with nonstationary input can be strongly influenced by the servicetime distribution beyond the mean. With sinusoidal arrival rates, the peak blocking probability typically increases as the servicetime distribution gets less variable. The influence of the servicetime distribution, including this seemingly anomalous behavior, can be understood and predicted from the modifiedofferedload and stationarypeakedness approximations, which exploit exact results for related infiniteserver models. Key Words: nonstationary queues; timedependent arrival rates; nonhomogeneous Markov chains; transient behavior; Erlang loss model; blocking probability; insensitivity; infiniteserver queues; modifiedofferedload approximation.
Control and recovery from rare congestion events in a large multiserver system
 Queueing Systems
, 1997
"... We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multiserver 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 ..."
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Cited by 16 (10 self)
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We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multiserver 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 multiserver 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 holdingtime distribution beyond its mean. The recovery time may or not be affected by the holdingtime 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 holdingtime distribution and the age distribution, focusing especially on the consequences of longtail distributions.
TwoParameter HeavyTraffic Limits for InfiniteServer Queues
"... Abstract: In order to obtain Markov heavytraffic approximations for infiniteserver queues with general nonexponential servicetime distributions and general arrival processes, possibly with timevarying arrival rates, we establish heavytraffic limits for twoparameter stochastic processes. We ..."
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Cited by 13 (8 self)
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Abstract: In order to obtain Markov heavytraffic approximations for infiniteserver queues with general nonexponential servicetime distributions and general arrival processes, possibly with timevarying arrival rates, we establish heavytraffic limits for twoparameter stochastic processes. We
A Nonstationary OfferedLoad Model for Packet Networks
, 1998
"... Motivated by the desire to model complex features of network traffic revealed in traffic measurements, such as heavytail probability distributions, longrange dependence, self similarity and nonstationarity, we propose a nonstationary offeredload model, in which connections of multiple types arriv ..."
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Cited by 12 (2 self)
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Motivated by the desire to model complex features of network traffic revealed in traffic measurements, such as heavytail probability distributions, longrange dependence, self similarity and nonstationarity, we propose a nonstationary offeredload model, in which connections of multiple types arrive according to independent nonhomogeneous Poisson processes, and general bandwidth stochastic processes describe the individual user bandwidth requirements at multiple links of a communication network during their connections. For example, an individual bandwidth process may be an onoff process where the on and off times have general (possibly heavytail) distributions. We obtain expressions for the moment generating function, mean and variance of the total required bandwidth of all customers on each link at any designated time. We suggest making decisions based on the probability that demand will exceed supply, or other designated target level, at each time of interest, using (i) numerical...
Variance reduction in simulation of loss models
 Operations Research
, 1999
"... We propose a new estimator of steadystate blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator base ..."
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Cited by 10 (8 self)
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We propose a new estimator of steadystate blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator based on the average number of customers in service, obtained from Little’s law (L = λW). It exploits the known offered load (product of the arrival rate and the mean service time). The variance reduction is dramatic when the blocking probability is high and the service times are highly variable. The advantage of the combination estimator in this regime is partly due to the indirect estimator, which itself is much more efficient than the natural estimator in this regime, and partly due to strong correlation (most often negative) between the natural and indirect estimators. In general, when the variances of two component estimators are very different, the variance reduction from the optimal convex combination is about 1 − ρ 2, where ρ is the correlation between the component estimators. For loss models, the variances of the natural and indirect estimators are very different under both light and heavy loads. The combination estimator is effective for estimating multiple blocking probabilities in loss networks with multiple traffic classes, some of which are in normal
A Network of TimeVarying ManyServer Fluid Queues with Customer Abandonment
"... To describe the congestion in largescale service systems, we introduce and analyze a nonMarkovian open network of manyserver fluid queues with customer abandonment, proportional routing and timevarying model elements. A proportion of the fluid completing service at each queue is routed immediate ..."
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Cited by 10 (10 self)
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To describe the congestion in largescale service systems, we introduce and analyze a nonMarkovian open network of manyserver fluid queues with customer abandonment, proportional routing and timevarying model elements. A proportion of the fluid completing service at each queue is routed immediately to each other queue, while the fluid not routed to other queues leaves the network. The fluid queue network serves as an approximation for the corresponding nonMarkovian open network of manyserver queues with Markovian routing, where all model elements may be time varying. We establish the existence of a unique vector of (net) arrival rate functions at each queue and the associated timevarying performance. In doing so, we provide the basis for an efficient algorithm, even for networks with many queues. Key words: queues with timevarying arrivals; queueing networks; manyserver queues; deterministic fluid model; customer abandonment; nonMarkovian queues. History: Submitted on February 7, 2010 1.
The Impact of Dependent Service Times on LargeScale Service Systems
"... This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic ..."
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Cited by 10 (9 self)
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This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic limit, allowing dependence among the service times. That limit shows that the number of customers in the system at any time is approximately Gaussian, where the timevarying mean is unaffected by the dependence, but the timevarying variance is affected by the dependence. As a consequence, required staffing to meet customary qualityofservice targets in a largescale service system with finitely many servers based on a normal approximation is primarily affected by dependence among the service times through this timevarying variance. This paper develops formulas and algorithms to quantify the impact of the dependence among the service times upon that variance. The approximation applies directly to infiniteserver models, but also indirectly to associated finiteserver models, exploiting approximations based on the peakedness (the ratio of the variance to the mean in the infiniteserver model). Comparisons with simulations confirm that the approximations can be useful to assess the impact of the dependence.