Adaptive server staffing in the presence of time-varying arrivals: a feed-forward control approach

We study the short-term staffing problem of systems that experience random, non-stationary demand. The typical method to accommodate changes in arrival rate is to use historical data to identify peak periods and associated forecasting for upcoming time windows. In this paper, we develop a method that instead detects change as it happens. Motivated by an automatic call distributor system in a call centre with time-varying arrivals, we propose a change detection algorithm based upon non-homogeneous Poisson processes. The proposed method is general and may be thought of as a feed-forward strategy, in which we detect a change in the arrival process, estimate the new magnitude of the arrival rate, and assign an appropriate number of servers to the tasks. The generalized likelihood ratio statistic is used and a recommendation for its decision limit is developed. Simulation results are used to evaluate the performance of the detector in the context of a telephone call centre.     

Analysis of call centre arrival data using singular value decomposition

We consider the general problem of analysing and modelling call centre arrival data. A method is described for analysing such data using singular value decomposition (SVD). We illustrate that the outcome from the SVD can be used for data visualization, detection of anomalies (outliers), and extraction of significant features from noisy data. The SVD can also be employed as a data reduction tool. Its application usually results in a parsimonious representation of the original data without losing much information. We describe how one can use the reduced data for some further, more formal statistical analysis. For example, a short‐term forecasting model for call volumes is developed, which is multiplicative with a time series component that depends on day of the week. We report empirical results from applying the proposed method to some real data collected at a call centre of a large‐scale U.S. financial organization. Some issues about forecasting call volumes are also discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Two Issues in Setting Call Centre Staffing Levels

Motivated by a problem facing the Police Communication Centre in Auckland, New Zealand, we consider the setting of staffing levels in a call centre with priority customers. The choice of staffing level over any particular time period (e.g., Monday from 8 am–9 am) relies on accurate arrival rate information. The usual method for identifying the arrival rate based on historical data can, in some cases, lead to considerable errors in performance estimates for a given staffing level. We explain why, identify three potential causes of the difficulty, and describe a method for detecting and addressing such a problem.     

The impact of special days in call arrivals forecasting: A neural network approach to modelling special days

A key challenge for call centres remains the forecasting of high frequency call arrivals collected in hourly or shorter time buckets. In addition to the complex intraday, intraweek and intrayear seasonal cycles, call arrival data typically contain a large number of anomalous days, driven by the occurrence of holidays, special events, promotional activities and system failures. This study evaluates the use of a variety of univariate time series forecasting methods for forecasting intraday call arrivals in the presence of such outliers. Apart from established, statistical methods, we consider artificial neural networks (ANNs). Based on the modelling flexibility of the latter, we introduce and evaluate different methods to encode the outlying periods. Using intraday arrival series from a call centre operated by one of Europe’s leading entertainment companies, we provide new insights on the impact of outliers on the performance of established forecasting methods. Results show that ANNs forecast call centre data accurately, and are capable of modelling complex outliers using relatively simple outlier modelling approaches. We argue that the relative complexity of ANNs over standard statistical models is offset by the simplicity of coding multiple and unknown effects during outlying periods.     

Optimizing Bernoulli Routing Policies for Balancing Loads on Call Centers and Minimizing Transmission Costs

We address the problem of assigning probabilities at discrete time instants for routing toll-free calls to a given set of call centers to minimize a weighted sum of transmission costs and load variability at the call centers during the next time interval.We model the problem as a tripartite graph and decompose the finding of an optimal probability assignment in the graph into the following problems: (i) estimating the true arrival rates at the nodes for the last time period; (ii) computing routing probabilities assuming that the estimates are correct. We use a simple approach for arrival rate estimation and solve the routing probability assignment by formulating it as a convex quadratic program and using the affine scaling algorithm to obtain an optimal solution.We further address a practical variant of the problem that involves changing routing probabilities associated with k nodes in the graph, where k is a prespecified number, to minimize the objective function. This involves deciding which k nodes to select for changing probabilities and determining the optimal value of the probabilities. We solve this problem using a heuristic that ranks all subsets of k nodes using gradient information around a given probability assignment.The routing model and the heuristic are evaluated for speed of computation of optimal probabilities and load balancing performance using a Monte Carlo simulation. Empirical results for load balancing are presented for a tripartite graph with 99 nodes and 17 call center gates.     

Robust and data-driven approaches to call centers

We propose both robust and data-driven approaches to a fluid model of call centers that incorporates random arrival rates with abandonment to determine staff levels and dynamic routing policies. We test the resulting models with real data obtained from the call center of a US bank. Computational results show that the robust fluid model is significantly more tractable as compared to the data-driven one and produces overall better solutions to call centers in most experiments.     

Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers

We consider a single server queueing system in which service shuts down when no customers are present, and is resumed when the queue length reaches a given critical length. We assume customers are heterogeneous on delay sensitivity and analyze customers’ strategic response to this mechanism and compare it to the overall optimal behavior. We provide algorithms to compute the equilibrium arrival rates and also derive the monotonicity of equilibrium and optimal arrival rates. We show that there may exist multiple equilibria in such a system and the optimal arrival rate may be larger or smaller than the decentralized equilibrium one.     

A Markov-modulated M/G/1 queue I: Stationary distribution

We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Asymptotic Analysis of a Loss Model with Trunk Reservation II: Trunks Reserved for Slow Traffic

We consider a model for a single link in a circuit switched network. The link has C circuits, and the input consists of offered calls of two types, that we call primary and secondary traffic. Of the C links R are reserved for primary traffic. We assume that both traffic types arrive as Poisson arrival streams. Assuming that C is large and   R = O (1)  , the arrival rate of secondary traffic is   O ( C )  while that of primary traffic is smaller, of the order     . The holding times of the secondary calls are assumed exponentially distributed with unit mean. Those of the primary calls are exponentially distributed with a large mean, that is     . The loads for both traffic types are thus comparable  ( O ( C ))  and we assume that the system is "critically loaded," i.e., the system's capacity is approximately equal to the total load. We analyze asymptotically the steady state probability that   n ₂  (resp.   n ₁  ) circuits are occupied by primary (resp. secondary) calls. In particular, we obtain two-term asymptotic approximations to the blocking probabilities for both traffic types. This work complements part I, where we assumed that the secondary traffic had an arrival rate that was     and a service rate that was     . Thus in part I the R trunks were reserved for the "fast traffic," whose arrival and service rates were   O ( C )  and   O (1)  .     

Performance analysis of an ISDN switch with distributed architecture: circuit switched calls

We have developed an approximate analytic model and a detailed simulation model to study the performance of an ISDN switch with distributed architecture. The analytic model treats the switch as a network of single server and infinite server queues with nonpreemptive priority service, general service times and batch arrivals. The simulation program is written in a distributed and modular way so as to simplify model development and debugging. Also extensive statistical techniques are employed for simulation output validation. It is observed that the analytic and the simulation models are in close agreement for the mean end-to-end delay and in moderately close agreement for the 95th percentile points of the end-to-end delay distribution. The comparisons between the analytic and the simulation models lead us to conjecture that the analytic model would be even more accurate for bigger systems with several hundred processors (where simulation models are too expensive to run). Even though the model assumes Poisson external call arrival process, it is shown that it may be applied with reasonable accuracy even when external call arrivals are non-Poisson. This is due to the fact that the composite message arrival process at a processor or transmission element tends to be close to Poisson even when the external call arrivals are non-Poisson.     

考虑运营稳定性的内支线班轮航线网络规划模型

货主选择承运航线的影响因素,既包括挂靠港口的计划到港时间与单箱运价,还包括反映班轮运营稳定性的甩箱率与准班率。对此,文章将挂靠港口的航行与在港时间不确定引入研究,并对挂靠港口间的不确定性建立联系,基于航次仿真来计算各挂靠港的到港时间分布、船舶的航次最大载箱量分布。以班轮航线的甩箱率与准班率限制、内支线最大船型与最长往返时间为约束,在优化内支线航线网络结构的同时,计算航线适配船型、班期密度及挂靠港计划到港时间。针对所构建的带不确定参数的NP难问题,文章设计了基于模拟仿真的智能优化算法,通过方案仿真技术来处理输入模型的众分布函数,借助智能优化原理从大范围解空间内寻找满意方案。文末对船舶航次仿真与网络规划模型的有效性进行了验证,算例分析表明:内支线班轮航线网络的货主选择比例达64%,且不论货主更偏好运输时间或价格,航线方案皆能贴近货主偏好。     

On the Counting Process for a Class of Markovian Arrival Processes with an Application to a Queueing System

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker–Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.AMS subject classification: 60K20, 60K25, 68M20This revised version was published online in June 2005 with corrected coverdate     

Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

Motivated by applications in telephone call centers, we consider a service system model with m customer classes and r server pools. The model is one with doubly stochastic arrivals, which means that the m-vector λ of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of dynamic control are considered: customers may be either blocked or accepted at the time of their arrival, and then accepted customers of each class must be routed, either immediately upon acceptance or after some period of waiting, to a server pool that is qualified to handle that class. Customers who are made to wait before commencement of their service are liable to defect. The objective is to minimize the expected sum of blocking costs, waiting costs and defection costs over a fixed and finite planning horizon. We consider an asymptotic parameter regime in which (i) the arrival rates, service rates and defection rates are uniformly accelerated by a large factor κ, then (ii) arrival rates are increased by an additional factor g(κ), and the number of servers in each pool is increased by g(κ) as well. This produces a separation of time scales, justifying a pointwise stationary stochastic fluid approximation for our original system model. In the stochastic fluid approximation, optimal admission control and routing decisions are determined by a simple linear program that uses the current arrival rate vector λ as data. We explain how to implement the fluid model's optimal control policy in our original service system context, and prove that the proposed implementation is asymptotically optimal in the first-order sense. AMS subject classification: 60K30, 90B15, 90B36     

Information security trade-offs and optimal patching policies

We develop and simulate a basic mathematical model of the costly deployment of software patches in the presence of trade-offs between confidentiality and availability. The model incorporates representations of the key aspects of the system architecture, the managers’ preferences, and the stochastic nature of the threat environment. Using the model, we compute the optimal frequencies for regular and irregular patching, for both networks and clients, for two example types of organization, military and financial. Such examples are characterized by their constellations of parameters. Military organizations, being relatively less cost-sensitive, tend to apply network patches upon their arrival. The relatively high cost of applying irregular client patches leads both types of organization to avoid deployment upon arrival.     

Non‐parametric modelling of time‐varying customer service times at a bank call centre

Call centres are becoming increasingly important in our modern commerce. We are interested in modelling the time‐varying pattern of average customer service times at a bank call centre. Understanding such a pattern is essential for efficient operation of a call centre. The call service times are shown to be lognormally distributed. Motivated by this observation and the important application, we propose a new method for inference about non‐parametric regression curves when the errors are lognormally distributed. Estimates and pointwise confidence bands are developed. The method builds upon the special relationship between the lognormal distribution and the normal distribution, and improves upon a naive estimation procedure that ignores this distributional structure. Our approach includes local non‐parametric estimation for both the mean function and the heteroscedastic variance function of the logged data, and uses local polynomial regression as a fitting tool. A simulation study is performed to illustrate the method. We then apply the method to model the time‐varying patterns of mean service times for different types of customer calls. Several operationally interesting findings are obtained and discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

American options and stochastic interest rates

We study finite-maturity American equity options in a stochastic mean-reverting diffusive interest rate framework. We allow for a non-zero correlation between the innovations driving the equity price and the interest rate. Importantly, we also allow for the interest rate to assume negative values, which is the case for some investment grade government bonds in Europe in recent years. In this setting we focus on American equity call and put options and characterize analytically their two-dimensional free boundary, i.e. the underlying equity and the interest rate values that trigger the optimal exercise of the option before maturity. We show that non-standard double continuation regions may appear, extending the findings documented in the literature in a constant interest rate framework. Moreover, we contribute by developing a bivariate discretization of the equity price and interest rate processes that converges in distribution as the time step shrinks. This discretization, described by a recombining quadrinomial tree, allows us to compute American equity options’ prices and to analyze their free boundaries with respect to time and current interest rate. Finally, we document the existence of non-standard optimal exercise policies for American call options on a non-dividend-paying equity.

     

A new model for call centre queue management

A new model for call centre queue management is described. It incorporates important features of call centre queues and is shown to produce results that are very different from those produced by the more usual models. The analytic approach is easy to apply, and is used to offer some interesting insights for call center queue management.     

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Individual customer’s behaviour in networks with state-dependent arrival rates

We consider Jackson networks with state-dependent arrival and service rates which show product form or nearly product form steady-states and come up with examples of load-dependent admission control. For these networks we prove an arrival theorem for external as well as for internal arrivals. In case of open tandem systems with state-independent service rates we compute the joint distribution of the sojourn times of a customer in the nodes and the distribution of the customer’s end-to-end-delay. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bounds for the tail distribution in a queue with a superposition of general periodic Markov sources: theory and application

An efficient yet accurate estimation of the tail distribution of the queue length has been considered as one of the most important issues in call admission and congestion controls in ATM networks. The arrival process in ATM networks is essentially a superposition of sources which are typically bursty and periodic either due to their origin or their periodic slot occupation after traffic shaping. In this paper, we consider a discrete-time queue where the arrival process is a superposition of general periodic Markov sources. The general periodic Markov source is rather general since it is assumed only to be irreducible, stationary and periodic. Note also that the source model can represent multiple time-scale correlations in arrivals. For this queue, we obtain upper and lower bounds for the asymptotic tail distribution of the queue length by bounding the asymptotic decay constant. The formulas can be applied to a queue having a huge number of states describing the arrival process. To show this, we consider an MPEG-like source which is a special case of general periodic Markov sources. The MPEG-like source has three time-scale correlations: peak rate, frame length and a group of pictures. We then apply our bound formulas to a queue with a superposition of MPEG-like sources, and provide some numerical examples to show the numerical feasibility of our bounds. Note that the number of states in a Markov chain describing the superposed arrival process is more than 1.4 × 10⁸⁸. Even for such a queue, the numerical examples show that the order of the magnitude of the tail distribution can be readily obtained.