Abstract not found

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 queue-ratio control with thresholds: When a pre-specified 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 heavy-traffic 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; efficiency-driven regime 1.

We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI many-server queueing model, which has a general time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time 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 time-varying 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 time-varying service capacity can be chosen to stabilize delays at any fixed target. Key words: queues with time-varying arrivals; nonstationary queues; many-server queues; deterministic fluid model; fluid approximation; queues with abandonment; non-Markovian queues.

Large-scale 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 system-wide service interruptions, as may occur with a system-wide 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 many-server heavy-traffic limits.

We develop analytical approximations to determine time-dependent staffing levels to stabilize abandonment probabilities and expected delays in the Mt/GI/st + GI many-server 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 modified-offered-load (MOL) approximations involving infinite-server models and show how they can be applied when the arrival rate is suitably high to determine a staffing function that makes the time-dependent 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; many-server queues; queues with time-varying arrivals; queues with abandonment; infinite-server queues, offered-load approximations; service systems; History: Submitted on June 6, 2009 1.

For high-value workforces in service organizations such as call centres, scheduling rules rely increasingly on queueing system models to achieve optimal performance. Most of these models assume a homogeneous population of servers, or at least a static service capacity per service agent. In this work we examine the challenge posed by dynamically fluctuating service capacity, where servers may increase their own service efficiency through experience; they may also decrease it through absence. We analyse the special case of a single agent selecting between two different job classes, and examine which of five service allocation policies performs best in the presence of learning and forgetting effects. We find that a type of specialisation minimises the steady state queue size; crosstraining boosts system capacity the most; and no simple policy matches a dynamic optimal cost policy under all conditions.

Staffing and scheduling optimization in large multiskill call centers is time-consuming, mainly because it requires lengthy simulations to evaluate performance measures and their sensitivity. Simplified models that provide tractable formulas are unrealistic in general. In this paper we explore an intermediate solution, based on an approximate continuous-time Markov chain model of the call center. This model is more accurate than the commonly used approximations, and yet can be simulated faster than a more realistic simulation (based on non-exponential distributions and additional details). To speed up the simulation, we uniformize the Markov chain and simulate only its discretetime version. We show how performance measures such as the fraction of calls of each type answered within a given waiting time limit can be recovered from this simulation, how to synchronize common random numbers in this setting, and how to use this in the first phase of an optimization algorithm based on the cutting plane method. We also discuss various implementation issues and provide empirical results. 1

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 single-period 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.

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 time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time 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 heavy-traffic limit confirm that this anomalous behavior also occurs in the large-scale queueing model.

To describe the congestion in large-scale service systems, we introduce and analyze a non-Markovian open network of many-server fluid queues with customer abandonment, proportional routing and time-varying 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 non-Markovian open network of many-server 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 time-varying performance. In doing so, we provide the basis for an efficient algorithm, even for networks with many queues. Key words: queues with time-varying arrivals; queueing networks; many-server queues; deterministic fluid model; customer abandonment; non-Markovian queues. History: Submitted on February 7, 2010, Revision submitted on July 28, 2010. 1.