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18
Choosing arrival process models for service systems: tests of a nonhomogeneous Poisson process.
 Nav. Res. Logist.
, 2014
"... Abstract: Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to per ..."
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Abstract: Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to perform statistical tests with service system data to confirm that an NHPP is actually appropriate, as emphasized by Brown et al.
A fluid approximation for the Gt/GI/st + GI queue
, 2010
"... We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general timevarying arrival process (the Gt), a general servicetime distribution (the first GI), a timedependent number of servers (the st) and allows ..."
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We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general timevarying arrival process (the Gt), a general servicetime distribution (the first GI), a timedependent number of servers (the st) and allows abandonment from queue according to a general abandonmenttime distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We characterize performance in the fluid model over alternating intervals in which the system is overloaded and underloaded (including critically loaded). For each t ≥ 0 and y ≥ 0, we determine the amount of fluid that is in service (in queue) at time t and has been so for time at most y. We obtain the service content density by applying the Banach contraction fixed point theorem. We also determine the timevarying potential waiting time, i.e., the virtual waiting time of a quantum of fluid arriving at a specified time, assuming that it will not abandon. The potential waiting time is determined by an ordinary differential equation. We show that a timevarying service capacity can be chosen to stabilize delays at any fixed target. Key words: queues with timevarying arrivals; nonstationary queues; manyserver queues; deterministic fluid model; fluid approximation; queues with abandonment; nonMarkovian queues.
STABILIZING PERFORMANCE IN NETWORKS OF QUEUES WITH TIMEVARYING ARRIVAL RATES
, 2014
"... This paper investigates extensions to feedforward queueing networks of an algorithm to set staffing levels (the number of servers) to stabilize performance in an Mt/GI/st + GI multiserver queue with a timevarying arrival rate. The model has a nonhomogeneous Poisson process (NHPP), customer aband ..."
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This paper investigates extensions to feedforward queueing networks of an algorithm to set staffing levels (the number of servers) to stabilize performance in an Mt/GI/st + GI multiserver queue with a timevarying arrival rate. The model has a nonhomogeneous Poisson process (NHPP), customer abandonment, and nonexponential service and patience distributions. For a single queue, simulation experiments showed that the algorithm successfully stabilizes abandonment probabilities and expected delays over a wide range of QualityofService (QoS) targets. A limit theorem showed that stable performance at fixed QoS targets is achieved asymptotically as the scale increases (by letting the arrival rate grow while holding the service and patience distributions fixed). Here we extend that limit theorem to a feedforward queueing network. However, these fixed QoS targets provide low QoS as the scale increases. Hence, these limits primarily support the algorithm with a low QoS target. For a high QoS target, effectiveness depends on the NHPP property, but the departure process never is exactly an NHPP. Thus, we investigate when a departure process can be regarded as approximately an NHPP. We show that index of dispersion for counts is effective for determining when a departure process is approximately an NHPP in this setting. In the important common case when all queues have high QoS targets, we show that both: (i) the departure process is approximately an NHPP from this perspective and (ii) the algorithm is effective.
Stabilizing Performance in a SingleServer Queue with TimeVarying Arrival Rate”. Working paper, Columbia University, available at: www.columbia.edu.
, 2014
"... Abstract We consider a general G t /G t /1 singleserver queue with unlimited waiting space and a timevarying arrival rate, where the the service rate at each time is subject to control. We first study the ratematching control, where the the service rate is made proportional to the arrival rate. W ..."
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Abstract We consider a general G t /G t /1 singleserver queue with unlimited waiting space and a timevarying arrival rate, where the the service rate at each time is subject to control. We first study the ratematching control, where the the service rate is made proportional to the arrival rate. We show that the model with the ratematching control can be regarded as a deterministic time transformation of a stationary G/G/1 model, so that the queue length distribution is stabilized as time evolves. However, the timevarying virtual waiting time is not stabilized. We show that the timevarying expected virtual waiting time with the ratematching servicerate control becomes inversely proportional to the arrival rate in a heavytraffic limit. We also show that no control that stabilizes the queue length asymptotically in heavytraffic can also stabilize the virtual waiting time. Then we consider a squareroot servicerate control, where the service rate exceeds the arrival rate by a constant multiple of the square root of the arrival rate. We show that this alternative servicerate control stabilizes the waiting time, but not the queue length, when the arrival rate changes very slowly relative to the average service time. This behavior is supported by a limit theorem supporting the pointwisestationary approximation.
A Markov model for measuring service levels in nonstationary G(t)/G(t)/s(t) + G(t) queues
"... A Markov model for measuring service levels in nonstationary ..."
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A Markov model for measuring service levels in nonstationary
Stabilizing Performance in a Service System with TimeVarying Arrivals and Customer Feedback
, 2015
"... Analytical approximations are developed to determine the timedependent offered load (effective demand) and appropriate staffing levels that stabilize performance at designated targets in a manyserver queueing model with timevarying arrival rates, customer abandonment from queue and random feedba ..."
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Analytical approximations are developed to determine the timedependent offered load (effective demand) and appropriate staffing levels that stabilize performance at designated targets in a manyserver queueing model with timevarying arrival rates, customer abandonment from queue and random feedback with additional delay after completing service. To provide a flexible model that can be readily fit to system data, the model has historydependent Bernoulli routing, where the feedback probabilities, servicetime and patience distributions all may depend on the visit number. Before returning to receive a new service, the fedback customers experience delays in an infiniteserver or finitecapacity queue, where the parameters may again depend on the visit number. A new refined modifiedofferedload approximation is developed to obtain good results with low waitingtime targets. Simulation experiments confirm that the approximations are effective. A manyserver heavytraffic FWLLN shows that the performance targets are achieved asymptotically as the scale increases.
Overload Control for a System in TimeVarying Environment
, 2013
"... In recent papers we considered how two large service systems that are primarily designed to operate independently, can help each other in face of unexpected overloads, due to a sudden change in the arrival rates. We proposed an overload control, which we named fixedqueueratio with thresholds (FQR ..."
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In recent papers we considered how two large service systems that are primarily designed to operate independently, can help each other in face of unexpected overloads, due to a sudden change in the arrival rates. We proposed an overload control, which we named fixedqueueratio with thresholds (FQRT), whose aim was to prevent any sharing of customers, i.e., sending customers from one class to be served in the other class ’ pool, during normal loads, and to initiate sharing automatically once a threshold is crossed, in which case the corresponding pool is considered overloaded. The goal is to keep the relation between the two queues fixed at a certain ratio, which is optimal in a deterministic “fluid” approximation, assuming a holding cost is incurred on the two queues. To avoid harmful sharing our control includes the oneway sharing rule, stipulating that sharing is allowed in only one direction at any time. In this paper we consider a more complex timevarying environment, in which the arrival rates and staffing levels are time dependent, so that the system may fluctuate between periods of various loads, with overloads possible in either direction. We show that FQRT needs to be modified to account for these more complex settings, since it may be slow to react to the changing environment, and may even cause sever fluctuations once the arrival rates return to normal after an overload incident. Our new control, FQR with activationandrelease thresholds (FQRART) is designed to automatically respond to changes in the environment by initiating sharing in the right direction quickly, if that is needed, while avoiding harmful phenomenons, such as congestion collapse and severe oscillations during normal loads. A novel fluid approximation, described implicitly via an ordinarydifferential equation (ODE) is developed, as well as an efficient algorithm to solve that ODE. 1
OM Forum Offered Load Analysis for Staffing
, 2013
"... This essay, based on my 2012 MSOM Fellow Lecture, discusses an idea that has been useful for developing effective methods to set staffing levels in service systems: offered load analysis. The main idea is to tackle a hard problem by first seeking an insightful simplification. For capacity planning t ..."
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This essay, based on my 2012 MSOM Fellow Lecture, discusses an idea that has been useful for developing effective methods to set staffing levels in service systems: offered load analysis. The main idea is to tackle a hard problem by first seeking an insightful simplification. For capacity planning to meet uncertain exogenous demand, offered load analysis looks at the amount of capacity that would be used if there were no constraints on its availability. This simplification is helpful because the stochastic model becomes much more tractable. Offered load analysis can be especially helpful when the demand is not only uncertain but also time varying, as in many service systems. Given the distribution of the stochastic offered load, we often can set staffing levels to stabilize performance at target levels, even in face of a strongly timevarying arrival rate, long service times, and network structure. Key words: offered load analysis; capacity planning; server staffing; timevarying arrival rates; infiniteserver queues
A Fluid Approximation for LargeScale Service Systems
"... We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general timevarying arrival process (the Gt), a general servicetime distribution (the first GI), a timedependent number of servers (the st) and allows ..."
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We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general timevarying arrival process (the Gt), a general servicetime distribution (the first GI), a timedependent number of servers (the st) and allows abandonment from queue according to a general abandonmenttime distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We also show that the system dynamics greatly simplifies in two special cases: (i) when the service time distribution is exponential (M) and (ii) when the service time distribution is deterministic (D) and the model is stationary. We develop an efficient algorithm to compute all standard performance functions in both cases. In case (i), we establish an asymptotic loss of memory (ALOM) property, i.e., asymptotic independence from the initial conditions as time evolves. We show that the difference in the performance functions with different initial conditions dissipates over time exponentially fast, under regularity conditions. In contrast, in case (ii) we show that ALOM fails dramatically. Instead, although all model parameters are constants, we show that the performance rapidly approaches a periodic steady state (PSS) with a period equal to the service time, whenever the system does not start with the unique stationary distribution. Moreover, the form of the PSS depends on the initial condition. Simulation and a heavytraffic limit confirm that this anomalous behavior also occurs in the largescale queueing model.