Probability and Measure

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Applied Probability and Queues.

Call centers are an increasingly important part of today's business world, employing millions of agents across the globe and serving as a primary customer-facing channel for firms in many different industries. Call centers have been a fertile area for operations management researchers in several domains, including forecasting, capacity planning, queueing, and personnel scheduling. In addition, as telecommunications and information technology have advanced over the past several years, the operational challenges faced by call center managers have become more complicated. Issues associated with human resources management, sales, and marketing have also become increasingly relevant to call center operations and associated academic research.

In this paper, we provide a survey of the recent literature on call center operations management. Along with traditional research areas, we pay special attention to new management challenges that have been caused by emerging technologies, to behavioral issues associated with both call center agents and customers, and to the interface between call center operations and sales and marketing. We identify a handful of broad themes for future investigation while also pointing out several very specific research opportunities.

/pdf/the-modern-call-center-a-multi-disciplinary-perspective-on-2kcrgkbqth.pdf

The Modern Call Center: A Multi-Disciplinary Perspective on Operations Management Research

/pdf/reversibility-and-stochastic-networks-17kl9p5ino.pdf

Reversibility and Stochastic Networks

Consider an aggregate arrival process A N obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τ on .A sN goes to infinity, with λN → � , A N approaches an M/G/∞ type process. Both for finite and infinite N , we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A ∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q P observed at the beginning of the arrival process activity periods ∞ x/(r+ρ−c) [τ on > u] d ux →∞ , where ρ = A ∞ < c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation. In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate et . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value et .

/pdf/asymptotic-results-for-multiplexing-subexponential-on-off-44g7yq1iy1.pdf

Asymptotic results for multiplexing subexponential on-off processes

Consider a finite list of items $n = 1, 2,\dots, N$, that are requested according to an i.i.d. process. Each time an item is requested it is moved to the front of the list. The associated search cost $C^N$ for accessing an item is equal to its position before being moved. If the request distribution converges to a proper distribution as $N \to \infty$, then the stationary search cost $C^N$ converges in distribution to a limiting search cost C. We show that, when the (limiting) request distribution has a heavy tail (e.g., generalized Zipf's law), $\mathbb{P}[R = n] \sim c/n^{\alpha}$ as $n \to \infty, \alpha > 1$, then the limiting stationary search cost distribution $\mathbb{P}[C > n]$, or, equivalently, the least-recently used (LRU) caching fault probability, satisfies $$\lim_{n \to \infty} \frac{\mathbb{P}[C > n]}{\mathbb{P}[R > n]}= (1 - \frac{1}{\alpha}) [\Gamma(1 - \frac{1}{\alpha})]^{\alpha} \nearrow e^{\gamma}$ \text{as}$\alpha \to \infty,$$ where $\Gamma$ is the Gamma function and $\gamma(=0.5772\dots)$ is Euler's constant. When the request distribution has a light tail $\mathbb{P}[R = n] \sim c \exp(-\lambdan_{\beta})$ as $n \to \infty (c, \lambda, \beta > 0), then $$\lim_{n \to \infty} \frac{\mathbb{P}[C_f > n]}{\mathbb{P}[R > n]} = e^{\gamma},$$ independently of $c, \lambda, \beta$, where $C_f$ is a fluid approximation of C. We experimentally demonstrate that the derived asymptotic formulas yield accurate results for lists of finite sizes. This should be contrasted with the exponential computational complexity of Burville and Kingman's exact expression for finite lists. The results also imply that the fault probability of LRU caching is asymptotically at most a factor $e^{\gamma} (\approx 1.78)$ greater than for the optimal static arrangement.

/pdf/asymptotic-approximation-of-the-move-to-front-search-cost-4vjh8yvck7.pdf

Asymptotic approximation of the move-to-front search cost distribution and least-recently used caching fault probabilities

Guided by the empirical observation that real-time MPEG video streams exhibit both multiple time scale and subexponential characteristics, we construct a video model that captures both of these characteristics and is amenable to queueing analysis. We investigate two fundamental approaches for extracting the model parameters: using sample path and second-order statistics-based methods. The model exhibits the following two canonical queueing behaviors. When strict stability conditions are satisfied, i.e., the conditional mean of each scene is smaller than the capacity of the server, precise modeling of the interscene dynamics (long-term dependency) is not essential for the accurate prediction of small to moderately large queue sizes. In this case, the queue length distribution is determined using quasistationary (perturbation theory) analysis. When weak stability conditions are satisfied, i.e., the conditional mean of at least one scene type is greater than the capacity of the server, the dominant effect for building a large queue size is the subexponential (long-tailed) scene length distribution. In this case, precise modeling of intrascene statistics is of secondary importance for predicting the large queueing behavior. A fluid model, whose arrival process is obtained from the video data by replacing scene statistics with their means, is shown to asymptotically converge to the exact queue distribution. Using the transition scenario of moving from one stability region to the other by a change in the value of the server capacity, we synthesize recent queueing theoretic advances and ad hoc results in video modeling, and unify a broad range of seemingly contradictory experimental observations found in the literature. As a word of caution for the widespread usage of second-order statistics modeling methods, we construct two processes with the same second-order statistics that produce distinctly different queueing behaviors.

https://www.ee.columbia.edu/%7Epredrag/mypub/JSACmpeg97.pdf

The effect of multiple time scales and subexponentiality in MPEG video streams on queueing behavior

Let {(Xn, Jn )} be a stationary Markov-modulated random walk on × E (E is finite), defined by its probability transition matrix measure F ={ F ij }, F ij (B) = [X1 ∈ B, J1 = j | J0 = i], B ∈ ( ), i, j ∈ E .I fF ij ([x, ∞))/(1 − H (x)) → W ij ∈ [0, ∞) ,a sx →∞ , for some long-tailed distribution function H , then the ascending ladder heights matrix distribution G+(x) (right Wiener–Hopf factor) has long-tailed asymptotics. If Xn 0, and H (x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by [supn≥0 Sn > x ]→ (− Xn ) −1 ∞ x [Xn > u] du as x →∞ ,w hereSn = n Xk , S0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov–modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI / GI /1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

/pdf/subexponential-asymptotics-of-a-markov-modulated-random-walk-1kknl89qek.pdf

Subexponential asymptotics of a markov-modulated random walk with queueing applications

https://www.ee.columbia.edu/%7Epredrag/mypub/JR02depLRU.pdf

Predrag R. Jelenković

Papers

Asymptotic results for multiplexing subexponential on-off processes

Asymptotic approximation of the move-to-front search cost distribution and least-recently used caching fault probabilities

The effect of multiple time scales and subexponentiality in MPEG video streams on queueing behavior

Subexponential asymptotics of a markov-modulated random walk with queueing applications

Least-recently-used caching with dependent requests