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Coping with TimeVarying Demand When Setting Staffing Requirements for a Service System
, 2007
"... We review queueingtheory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the longrun steadystate behavior of stationary mode ..."
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Cited by 40 (16 self)
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We review queueingtheory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the longrun steadystate behavior of stationary models. We show how to adapt stationary queueing models for use in nonstationary environments so that timedependent performance is captured and staffing requirements can be set. Relatively little modification of straightforward stationary analysis applies in systems where service times are short and the targeted quality of service is high. When service times are moderate and the targeted quality of service is still high, timelag refinements can improve traditional stationary independent periodbyperiod and peakhour approximations. Timevarying infiniteserver models help develop refinements, because closedform expressions exist for their timedependent behavior. More difficult cases with very long service times and other complicated features, such as endofday effects, can often be treated by a modifiedofferedload approximation, which is based on an associated infiniteserver model. Numerical algorithms and deterministic fluid models are useful when the system is overloaded for an extensive period of time. Our discussion focuses on telephone call centers, but applications to police patrol, banking, and hospital emergency rooms are also mentioned.
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 20 (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.
Uniform acceleration expansions for Markov chains with timevarying rates
 Annals of Applied Probability
, 1997
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A survey and experimental comparison of service level approximation methods for nonstationary M/M/s queueing systems
 INFORMS Journal of Computing
, 2005
"... We compare the performance of six methods in computing or approximating service levels for nonstationary M/M/s queueing systems: an exact method (a Runge Kutta ordinary differential equation solver), the randomization method, a closure (or surrogate distribution) approximation, a direct infinite ser ..."
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Cited by 14 (2 self)
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We compare the performance of six methods in computing or approximating service levels for nonstationary M/M/s queueing systems: an exact method (a Runge Kutta ordinary differential equation solver), the randomization method, a closure (or surrogate distribution) approximation, a direct infinite server approximation, a modified offered load infinite server approximation, and an effective arrival rate approximation. We used all of the methods to solve the same set of 128 test problems. The randomization method was almost as accurate as the exact method, and used less than half the computational time of the exact method. The closure approximation was less accurate, and in many cases slower, than the randomization method. The two infinite server based approximations and the effective arrival rate approximation had were less accurate but had computation times that were far shorter and less problemdependent than for the other three methods.
Unstable asymptotics for nonstationary queues
 Math. Oper. Res
, 1994
"... We relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes. We apply these results to develop approximations for queues that are unstable in a nonstationary manner. We obtain unstable nonstationary analogs of th ..."
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Cited by 10 (7 self)
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We relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes. We apply these results to develop approximations for queues that are unstable in a nonstationary manner. We obtain unstable nonstationary analogs of the queueing relation L = λW and associated centrallimittheorem versions. For modeling and to obtain the first limits, we can construct nonstationary point processes as random timetransformations of familiar point processes, such as renewal processes and stationary point processes. We deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation.
Realtime delay estimation based on delay history
, 2007
"... 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 realtime delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the ..."
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Cited by 9 (4 self)
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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 realtime 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 delayhistory 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 multiserver 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 manyserver and classical heavytraffic limiting regimes.
WaitTime Predictors for Customer Service Systems With TimeVarying Demand and Capacity
"... We develop new improved realtime delay predictors for manyserver service systems with a timevarying arrival rate, a timevarying number of servers and customer abandonment. We develop four new predictors, two of which exploit an established deterministic fluid approximation for a manyserver queu ..."
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Cited by 7 (1 self)
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We develop new improved realtime delay predictors for manyserver service systems with a timevarying arrival rate, a timevarying number of servers and customer abandonment. We develop four new predictors, two of which exploit an established deterministic fluid approximation for a manyserver queueing model with those features. These delay predictors may be used to make delay announcements. We use computer simulation to show that the proposed predictors outperform previous predictors.
Largetime asymptotics for the Gt/Mt/st + GIt manyserver fluid queue with customer abandonment
, 2010
"... We previously introduced and analyzed the Gt/Mt/st +GIt manyserver fluid queue with timevarying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper we establish an asympt ..."
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Cited by 6 (6 self)
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We previously introduced and analyzed the Gt/Mt/st +GIt manyserver fluid queue with timevarying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper we establish an asymptotic loss of memory (ALOM) property for that fluid model; i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s + GI fluid queue converges to steady state and the periodic Gt/Mt/st + GIt fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.
Limits For Queues As The Waiting Room Grows
 Queueing Systems
, 1989
"... We study the convergence of finitecapacity open queueing systems to their infinitecapacity counterparts as the capacity increases. Convergence of the transient behavior is easily established in great generality provided that the finitecapacity system can be identified with the infinitecapacity sy ..."
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Cited by 4 (0 self)
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We study the convergence of finitecapacity open queueing systems to their infinitecapacity counterparts as the capacity increases. Convergence of the transient behavior is easily established in great generality provided that the finitecapacity system can be identified with the infinitecapacity system up to the first time that the capacity is exceeded. Convergence of steadystate distribution is more difficult; it is established here for singlefacility models such as GI/GI/c/n with c servers, n  c extra waiting space and the firstcome firstserved discipline, in which all arrivals finding the waiting room full are lost without affecting future arrivals, via stability properties of generalized semiMarkov processes. 1. Introduction Consider an open queueing system with capacity n. When n is very large, we expect that the standard descriptive stochastic processes, such as the number of customers in the system at time t for t 0, and their limiting steadystate distributions are ve...
Stationary birthanddeath processes fit to queues with periodic arrival rate functions. Working paper
, 2015
"... To better understand how to interpret birthanddeath (BD) processes fit to service system data, we investigate the consequences of fitting a BD process to a multiserver queue with a periodic timevarying arrival rate function. We study how this fitted BD process is related to the original queuele ..."
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Cited by 2 (2 self)
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To better understand how to interpret birthanddeath (BD) processes fit to service system data, we investigate the consequences of fitting a BD process to a multiserver queue with a periodic timevarying arrival rate function. We study how this fitted BD process is related to the original queuelength process. If a BD process is fit to a segment of the sample path of the queuelength process, with the birth (death) rates in each state estimated by the observed number of arrivals (departures) in that state divided by the total time spent in that state, then under minor regularity conditions that BD process has the steadystate distribution of the queue length process in the original Mt/GI/s queueing model as the sample size increases. The steadystate distribution can be estimated efficiently by fitting a parametric function to the observed birth and death rates.