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51
A Fluid Approximation for Service Systems Responding to Unexpected Overloads
"... In a recent paper we considered two networked service systems, each having its own customers and designated service pool with many agents, where all agents are able to serve the other customers, although they may do so inefficiently. Usually we want the agents to serve only their own customers, but ..."
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Cited by 12 (9 self)
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In a recent paper we considered two networked service systems, each having its own customers and designated service pool with many agents, where all agents are able to serve the other customers, although they may do so inefficiently. Usually we want the agents to serve only their own customers, but we want an automatic control that activates serving some of the other customers when an unexpected overload occurs. Assuming that we will not know the timing or the extent of the overload, we proposed a queueratio control with thresholds: When a prespecified threshold is crossed, serving the other customers is activated, so that a certain queue ratio, which is determined by only observing the queue lengths, is maintained. We then developed a simple fluid approximation in which this control was shown to be optimal, and we showed how to calculate the control parameters. In this paper, we focus on the fluid approximation itself, and develop its transient equations, which depend on a heavytraffic averaging principle. Although deterministic, the new fluid model includes stochastic balance equations. The full fluid model here enables us to analyze scenarios we could not analyze previously, e.g., when the thresholds are small, and are being crossed over frequently. We also use this new fluid approximation to refine our previous results. Simulation experiments show that the refined fluid model greatly increases the accuracy of our performance predictions. Key words: service systems; unexpected overloads; fluid approximation; averaging principle; efficiencydriven regime 1.
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.
Steadystate analysis of a multiserver queue in the HalfinWhitt regime
, 2008
"... We examine a multiserver queue in the HalfinWhitt (Quality and EfficiencyDriven) regime: as the number of servers n increases, the utilization approaches 1 from below at the rate Θ(1 / √ n). The arrival process is renewal and service times have a latticevalued distribution with a finite suppor ..."
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Cited by 8 (0 self)
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We examine a multiserver queue in the HalfinWhitt (Quality and EfficiencyDriven) regime: as the number of servers n increases, the utilization approaches 1 from below at the rate Θ(1 / √ n). The arrival process is renewal and service times have a latticevalued distribution with a finite support. We consider the steadystate distribution of the queue length and waiting time in the limit as the number of servers n increases indefinitely. The queue length distribution, in the limit as n → ∞, is characterized in terms of the stationary distribution of an explicitly constructed Markov chain. As a consequence, the steadystate queue length and waiting time scale as Θ ( √ n) and Θ(1 / √ n) as n → ∞, respectively. Moreover, an explicit expression for the critical exponent is derived for the moment generating function of a limiting (scaled) steadystate queue length. This exponent depends on three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a singleserver queue in the conventional heavytraffic regime. The results are derived by analyzing Lyapunov functions.
Stabilizing Customer Abandonment in ManyServer Queues with TimeVarying Arrivals
"... We develop analytical approximations to determine timedependent staffing levels to stabilize abandonment probabilities and expected delays in the Mt/GI/st + GI manyserver queueing model, which has a nonhomogeneous Poisson arrival process (the Mt), general service times (the first GI) and allows cu ..."
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Cited by 6 (4 self)
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We develop analytical approximations to determine timedependent staffing levels to stabilize abandonment probabilities and expected delays in the Mt/GI/st + GI manyserver queueing model, which has a nonhomogeneous Poisson arrival process (the Mt), general service times (the first GI) and allows customer abandonment according to a general patience distribution (the +GI). In particular, we propose new offeredload (OL) and modifiedofferedload (MOL) approximations involving infiniteserver models and show how they can be applied when the arrival rate is suitably high to determine a staffing function that makes the timedependent abandonment probability and expected potential waiting time (the virtual waiting time of a customer with infinite patience) nearly constant over time at targeted levels, after an initial transient. We also develop approximations for other performance measures under this staffing function. These approximations show that the other performance measures are not stabilized to the same extent, and quantify their fluctuations. We perform simulations to show that the approximations are effective. Key words: staffing; capacity planning; manyserver queues; queues with timevarying arrivals; queues with abandonment; infiniteserver queues, offeredload approximations; service systems; History: Submitted on June 6, 2009 1.
Service Interruptions In LargeScale Service Systems
"... Largescale service systems, where many servers respond to high demand, are appealing because they can provide great economy of scale, producing a high quality of service with high efficiency. Customer waiting times can be short, with a majority of customers served immediately upon arrival, while se ..."
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Cited by 4 (2 self)
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Largescale service systems, where many servers respond to high demand, are appealing because they can provide great economy of scale, producing a high quality of service with high efficiency. Customer waiting times can be short, with a majority of customers served immediately upon arrival, while server utilizations remain close to 100%. However, we show that this confluence of quality and efficiency is not achieved without risk, because there can be severe congestion if the system does not operate as planned. In particular, we show that the large scale makes the system more vulnerable to systemwide service interruptions, as may occur with a systemwide computer failure. Increasing scale can lead to longer recovery times from service interruptions and worse performance during such events. We quantify the impact of service interruptions with increasing scale by introducing and analyzing queueing models. We apply approximating deterministic fluid models, which can be obtained from manyserver heavytraffic limits.
Choosing Arrival Process Models for Service Systems: Tests of a Nonhomogeneous Poisson Process
, 2013
"... 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 statis ..."
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Cited by 3 (3 self)
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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. [6]. They suggested a specific statistical test based on the KolmogorovSmirnov statistic after exploiting the conditionaluniform property to transform the NHPP into a sequence of i.i.d. random variables uniformly distributed on [0, 1] and then performing a logarithmic transformation of the data. We conduct extensive simulation experiments to study the power of that statistical test and various alternatives. We conclude that the general approach of Brown et al. [6] is excellent, but that an alternative KolmogorovSmirnov test proposed by Lewis [15], exploiting a different transformation due to Durbin [7], consistently has greater power.
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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Cited by 3 (3 self)
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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.
Critically loaded queueing models that are throughput suboptimal
 Ann. Appl. Probab
, 2009
"... This paper introduces and analyzes the notion of throughput suboptimality for manyserver queueing systems in heavy traffic. The queueing model under consideration has multiple customer classes, indexed by a finite set I, and heterogenous, exponential servers. Servers are dynamically chosen to serve ..."
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Cited by 1 (1 self)
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This paper introduces and analyzes the notion of throughput suboptimality for manyserver queueing systems in heavy traffic. The queueing model under consideration has multiple customer classes, indexed by a finite set I, and heterogenous, exponential servers. Servers are dynamically chosen to serve customers, and buffers are available for customers waiting to be served. The arrival rates and the number of servers are scaled up in such a way that the processes representing the number of classi customers in the system, i ∈ I, fluctuate about a static fluid model, that is assumed to be critically loaded in a standard sense. At the same time, the fluid model is assumed to be throughput suboptimal. Roughly, this means that the servers can be allocated so as to achieve a total processing rate that is greater than the total arrival rate. We show that there exists a dynamic control policy for the queueing model that is efficient in the following strong sense: Under this policy, for every finite T, the measure of the set of times prior to T, at which at least one customer is in the buffer, converges to zero in probability as the arrival rates and number of servers go to infinity. On the way to prove our main result, we provide a characterization of throughput suboptimality in terms of properties of the bufferstation graph.
“Nursevendor Problem”: Personnel Staffing in the Presence of Endogenous Absenteeism
, 2010
"... The problem of determining nurse staffing levels in a hospital environment is a complex task due to variable patient census levels and uncertain service capacity caused by nurse absenteeism. In this paper, we combine an empirical investigation of the factors affecting nurse absenteeism rates with an ..."
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Cited by 1 (0 self)
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The problem of determining nurse staffing levels in a hospital environment is a complex task due to variable patient census levels and uncertain service capacity caused by nurse absenteeism. In this paper, we combine an empirical investigation of the factors affecting nurse absenteeism rates with an analytical treatment of nurse staffing decisions using a novel variant of the newsvendor model. Using data from the emergency department of a large urban hospital, we find that absenteeism rates are correlated with anticipated future nurse workload levels. Using our empirical findings, we analyze a singleperiod nurse staffing problem considering both the case of constant absenteeism rate (exogenous absenteeism) as well as an absenteeism rate which is a function of the number of scheduled nurses (endogenous absenteeism). We provide characterizations of the optimal staffing levels in both situations and show that the failure to incorporate absenteeism as an endogenous effect results in understaffing.