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11
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
Staffing a Call Center with Uncertain Arrival Rate and Absenteeism
 Production and Operations Management
"... This paper proposes simple methods for staffing a singleclass call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multiserver queue with customer aba ..."
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Cited by 22 (4 self)
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This paper proposes simple methods for staffing a singleclass call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multiserver queue with customer abandonment, allowing nonexponential servicetime and timetoabandon distributions. The goal is to maximize the expected net return, given throughput benefit and server, customerabandonment and customerwaiting costs, but attention is also given to the standard deviation of the return. The approach is to approximate the performance and the net return, conditional on the random modelparameter vector, and then uncondition to get the desired results. Two recentlydeveloped approximations are used for the conditional performance measures: first, a deterministic fluid approximation and, second, a numerical algorithm based on a purely Markovian birthanddeath model, having statedependent death rates. Key words: modelparameter uncertainty; contact centers; employee absenteeism; customer abandonment; fluid models
Decomposition Approximation for timedependent Markovian queueing networks
 Operations Research Letters
, 1999
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Twotimescale markov chains and applications to quasibirthdeath queues
 SIAM journal on applied mathematics
, 2005
"... Abstract. Aiming at reduction of complexity, this work is concerned with twotimescale Markov chains and applications to quasibirthdeath queues. Asymptotic expansions of probability vectors are constructed and justified. Lumping all states of the Markov chain in each subspace into a single state, ..."
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Cited by 2 (1 self)
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Abstract. Aiming at reduction of complexity, this work is concerned with twotimescale Markov chains and applications to quasibirthdeath queues. Asymptotic expansions of probability vectors are constructed and justified. Lumping all states of the Markov chain in each subspace into a single state, an aggregated process is shown to converge to a continuoustime Markov chain whose generator is an average with respect to the stationary measures. Then a suitably scaled sequence is shown to converge to a switching diffusion process. Extensions of the results are presented together with examples of quasibirthdeath queues. Key words. Markov chain, singular perturbation, countable state space, asymptotic expansion, occupation measure, aggregation, switching diffusion, quasibirthdeath queue
A Fluid Limit for an Overloaded X Model Via an Averaging Principle
, 2010
"... We prove a manyserver heavytraffic fluid limit for an overloaded Markovian queueing system having two customer classes and two service pools, known in the callcenter literature as the X model. The system uses the fixedqueueratiowiththresholds (FQRT) control, which we proposed in a recent pape ..."
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Cited by 2 (2 self)
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We prove a manyserver heavytraffic fluid limit for an overloaded Markovian queueing system having two customer classes and two service pools, known in the callcenter literature as the X model. The system uses the fixedqueueratiowiththresholds (FQRT) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQRT, customers are served by their own service pool until a threshold is exceeded. Then, oneway sharing is activated with customers from one class allowed to be served in both pools. After the control is activated, it aims to keep the two queues at a prespecified fixed ratio. For large systems that fixed ratio is achieved approximately. For the fluid limit, or FWLLN, we consider a sequence of properly scaled X models in overload operating under FQRT. Our proof of the FWLLN follows the compactness approach, i.e., we show that the sequence of scaled processes is tight, and then show that all converging subsequences have the specified limit. The characterization step is complicated because the queuedifference processes, which determine the customerserver assignments, remain stochastically bounded, and need to be considered without spatial scaling. Asymptotically, these queuedifference processes operate in a faster time scale than the fluidscaled processes. In the limit, due to a separation of time scales, the driving processes converge to a timedependent steady state (or local average) of a timevarying fasttimescale process (FTSP). This averaging principle (AP) allows us to replace the driving processes with the longrun average behavior of
What You Should Know About Queueing Models To Set Staffing Requirements in Service Systems by
, 2007
"... One traditional application of queueing models is to help set staffing requirements in service systems, but the way to do so is not entirely straightforward, largely because demand in service systems typically varies greatly by the time of day. This paper discusses ways old and newto cope with tha ..."
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One traditional application of queueing models is to help set staffing requirements in service systems, but the way to do so is not entirely straightforward, largely because demand in service systems typically varies greatly by the time of day. This paper discusses ways old and newto cope with that timevarying demand.
with timevarying rates and redialing subscribers ∗
, 2001
"... www.math.utwente.nl/publications ..."
SINGULARLY PERTURBED MARKOV CHAINS: LIMIT RESULTS AND APPLICATIONS
, 2007
"... This work focuses on timeinhomogeneous Markov chains with two time scales. Our motivations stem from applications in reliability and dependability, queueing networks, financial engineering and manufacturing systems, where twotimescale scenarios naturally arise. One of the important questions is: ..."
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This work focuses on timeinhomogeneous Markov chains with two time scales. Our motivations stem from applications in reliability and dependability, queueing networks, financial engineering and manufacturing systems, where twotimescale scenarios naturally arise. One of the important questions is: As the rate of fluctuation of the Markov chain goes to infinity, if the limit distributions of suitably centered and scaled sequences of occupation measures exist, what can be said about the convergence rate? By combining singular perturbation techniques and probabilistic methods, this paper addresses the issue by concentrating on sequences of centered and scaled functional occupation processes. The results obtained are then applied to treat a queueing system example. 1. Introduction. This
A Fluid Limit for an Overloaded X Model Via a Stochastic Averaging Principle
"... Weproveamanyserverheavytrafficfluidlimit for anoverloadedMarkovianqueueingsystemhavingtwocustomer classes and two service pools, known in the callcenter literature as the X model. The system uses the fixedqueueratiowiththresholds (FQRT) control, which we proposed in a recent paper as a way fo ..."
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Weproveamanyserverheavytrafficfluidlimit for anoverloadedMarkovianqueueingsystemhavingtwocustomer classes and two service pools, known in the callcenter literature as the X model. The system uses the fixedqueueratiowiththresholds (FQRT) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQRT, customers are served by their own service pool until a threshold is exceeded. Then, oneway sharing is activated with customers from one class allowed to be served in both pools. After the control is activated, it aims to keep the two queues at a prespecified fixed ratio. For large systems that fixed ratio is achieved approximately. For the fluid limit, or FWLLN, we consider a sequence of properly scaled X models in overload operating under FQRT. Our proof of the FWLLN follows the compactness approach, i.e., we show that the sequence of scaled processes is tight, and then show that all converging subsequences have the specified limit. The characterization step is complicated because the queuedifference processes, which determine the customerserver assignments, need to be considered without spatial scaling. Asymptotically, these queuedifference processes operate on a faster time scale than the fluidscaled processes. In the limit, due to a separation of time scales, the driving processes converge to a timedependent steady state (or local average) of a timevarying fasttimescale process (FTSP). This averaging principle allows us to replace the driving processes with the longrun average behavior of the FTSP.
Strong approximations for Markovian service networks
, 1997
"... Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the numbe ..."
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Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of Mt/Mt/nt queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “nonsmooth ” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this nonsmoothness by using a new notion of derivative.