Abstract
We consider a controlled queueing system of the \(G/M/n/B+GI\) type, with many servers and impatient customers. The queue-capacity \(B\) is the control process. Customers who arrive at a full queue are blocked and customers who wait too long in the queue abandon. We study the tradeoff between blocking and abandonment, with cost accumulated over a random, finite time-horizon, which yields a queueing control problem (QCP). In the many-server quality and efficiency-driven (QED) regime, we formulate and solve a diffusion control problem (DCP) that is associated with our QCP. The DCP solution is then used to construct asymptotically optimal controls (of the threshold type) for QCP. A natural motivation for our QCP is telephone call centers, hence the QED regime is natural as well. QCP then captures the tradeoff between busy signals and customer abandonment, and our solution specifies an asymptotically optimal number of trunk-lines.
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References
Atar, R.: Scheduling control for queueing systems with many servers: asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15, 2606–2650 (2005)
Atar, R., Mandelbaum, A., Reiman, M.I.: Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic. Ann. Appl. Probab. 14, 1084–1134 (2004)
Borodin, A.N., Salminen, P.: Handbook of Brownian Motion-Facts and Formulae, 2nd edn. Birkhäuser-Verlag, Boston (2002)
Burdzy, K., Kang, W., Ramanan, K.: The Skorokhod problem in a time-dependent interval. Stoch. Proc. Appl. 119, 428–452 (2009)
Borst, S., Mandelbaum, A., Reiman, M.I.: Dimensioning large call centers. Oper. Res. 52, 17–34 (2004)
Dai, J.G., He, S.: Customer abandonment in many-server queues. Math. Oper. Res. 35, 347–362 (2010)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2006)
Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review and research prospects. Manuf Serv. Oper. Manag. 5, 79–141 (2003)
Garnett, O., Mandelbaum, A., Reiman, M.I.: Designing a telephone call center with impatient customers. Manuf. Serv. Oper. Manag. 4, 208–227 (2002)
Ghosh, A.P., Weerasinghe, A.P.: Optimal buffer size for a stochastic processing network in heavy traffic. Queueing Syst. 55, 147–159 (2007)
Ghosh, A.P., Weerasinghe, A.P.: ptimal buffer size and dynamic rate control for a queueing network in heavy traffic. Stoch. Proc. Appl. 120, 2103–2141 (2010)
Halfin, S., Whitt, W.: Heavy traffic limits for queues with many exponential servers. Oper. Res. 29, 567–588 (1981)
Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Wiley Publications, New York (1985)
Hartman, P.: Ordinary Differential Equations. Wiley Publications, New York (1964)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland Publishers, Amsterdam (1981)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)
Khudyakov, P.: Designing a Call Center with an IVR (Interactive Voice Response), Technion M.Sc. Thesis (2006). Downloadable at http://iew3.technion.ac.il/serveng/References/thesis_polyna.pdf
Khudyakov, P., Feigin, P., Mandelbaum, A.: Designing a call center with an interactive voice response (IVR). Queueing Syst. 66, 215–237 (2010)
Kocaga, Y.L., Ward, A.: Admission control for a multi-server queue with abandonment. Queueing Syst. 65, 275–323 (2010)
Krichagina, E.V., Taksar, M.: Diffusion approximation for GI/G/1 controlled queues. Queueing Syst. Theory Appl. 12, 333–367 (1992)
Kruk, L., Lehoczky, J., Ramanan, K., Shreve, S.: An explicit formula for the Skorokhod map on [0, a]. Ann. Appl. Probab. pp. 669–682 (2007)
Kumar, S., Muthuraman, M.: A numerical method for solving singular stochastic control problems. Oper. Res. 52, 563–582 (2004)
Lee, C., Weerasinghe, A.: Convergence of a queueing system in heavy traffic with general patience-time distributions. Stoch. Proc. Appl. 121, 2507–2552 (2011)
Ma, J.: On the principle of smooth-fit for a class of singular stochastic control problems for diffusions. SIAM J. Control Opt. 30, 975–999 (1992)
Mandelbaum, A., Momcilovic, P.: Queues with many servers and impatient customers. Math. Oper. Res. 37, 41–64 (2012)
Massey, A.W., Wallace, B.R.: An optimal design of the M/M/C/K queue for call centers, To appear in QUESTA
Meyer, P.A.: Un cours sur les integrales stochastiques, Seminaire de Probabilities X. Lecture Notes in Mathematics, vol. 511. Springer, New York (1974)
Pang, G., Talreja, R., Whitt, W.: Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4, 193–267 (2007)
Protter, P.: Stochastic Differential Equations, 2nd edn. Springer, Berlin (2004)
Puhalskii, A.A., Reiman, M.I.: The multi-class \(GI/PH/N \) queue in the Halfin–Whitt regime. Adv. Appl. Probab. 32, 564–595 (2000)
Reed, J.E.: The \(G/GI/N\) queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19, 2211–2269 (2009)
Reed, J.E., Ward, A.R.: Approximating the \(GI/GI/1+GI\) queue with a nonlinear drift diffusion: hazard rate scaling in heavy traffic. Math. Oper. Res. 33, 606–644 (2008)
van Leeuwaarden, J., Janssen, A.J.E.M., Zwart, B.: Gaussian expansions and bounds for the Poisson distribution with application to the Erlang B formula. Adv. Appl. Probab. 40, 122–143 (2008)
van Leeuwaarden, J., Janssen, A.J.E.M., Zwart, B.: Refining square root safety staffing by expanding Erlang C. Oper. Res. 59, 1512–1522 (2011)
Ward, A., Kumar, S.: Asymptotically optimal admission control of a queue with impatient customers. Math. Oper. Res. 33, 167–202 (2008)
Whitt, W.: Heavy traffic limits for the \(G/H^*_2/n/m \) queue. Math. Oper. Res. 30, 1–27 (2005)
Weerasinghe, A.: A bounded variation control problem for diffusion processes. SIAM J. Control Opt. 44, 389–417 (2005)
Acknowledgments
The research of A.W. has been supported in part by the Army Research Office under Grant No. W 911NF0710424. The work of A.M. has been partially supported by BSF Grants 2005175 and 2008480, ISF Grant 1357/08 and by the Technion funds for promotion of research and sponsored research. Some of the research was funded by and carried out while A.M. was visiting the Statistics and Applied Mathematical Sciences Institute (SAMSI) of the NSF; the Department of Statistics and Operations Research (STOR), the University of North Carolina at Chapel Hill; the Department of Information, Operations and Management Sciences (IOMS), Leonard N. Stern School of Business, New York University; and the Department of Statistics, The Wharton School, University of Pennsylvania—the wonderful hospitality of all four institutions is gratefully acknowledged and truly appreciated. We would also like to thank former graduate student Ju Ming, at the Mathematics Department of Iowa State University, for his help in preparing a Matlab program for the numerical computations in Sect. 3.4. We are grateful to Lillian Bluestein, of the Faculty of Industrial Engineering and Management, Technion, for her technical assistance in preparation of this manuscript. Finally, we thank the referees for valuable suggestions that has led to a significantly improved manuscript.
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Appendix
Appendix
1.1 Proof of Lemma 3.2
Let \(x \le 0\) and consider the diffusion process \(Y\) characterized by
where \(W\) is a standard Brownian motion. Next, we introduce the stopping time \(\tau _0\) by
We intend to show that the function \(F_\infty \) of (3.15) and (3.16) has the stochastic representation \( F_\infty (x)=E[\mathrm{{e}}^{ -\gamma \tau _0}| Y(0)=x]\), for every \(x \le 0\). To this end, introduce
for all \(x \le 0\). For each \(N \ge |x|\), we also introduce the stopping time
To construct the function \(F_\infty \), we begin with a sequence of functions \((F_n)\). Let \(F_n\) be the unique solution to the boundary value problem
Then, with the use of It\(\hat{o}\)’s lemma, one can easily verify that
Using the scale function associated with the diffusion \(Y\), it follows that (see [17], Sect. 4.1 of Chapter 5)
Therefore, \(P[ \tau _n < \tau _0 | Y(0)=x]\) is decreasing to zero as \(n\) tends to infinity. Consequently, the sequence \((F_n(x))\) is increasing to the function \(\tilde{F}_\infty \). We fix the interval \([x,0]\) and integrate the differential equation for \(F_n\) twice on this interval to obtain an integral equation for \(F_n\). Then we use \(0\le F_n(y) \le \tilde{F}_\infty (y) \le 1\), on \([-n,0]\), and \(\lim \limits _{n \rightarrow \infty }F_n(y)=\tilde{F}_\infty (y)\), together with the bounded convergence theorem, to conclude that the function \(\tilde{F}_\infty \) also satisfies the same integral equation. By differentiating it twice, we observe that \(\tilde{F}_\infty \) also satisfies (3.15), together with the boundary condition \(\tilde{F}_\infty (0)=1\).
The stochastic representation (5.3) also implies that \(\tilde{F}_\infty \) is increasing and \(\tilde{F}^{\prime }_\infty (x) \ge 0\) on the interval \((-\infty ,0]\). Consequently, \(\lim \limits _{x \rightarrow -\infty } \tilde{F}_\infty (x) =L_0\) exists, with \(0 \le L_0 <1\). Furthermore, if \(\tilde{F}^{\prime }_\infty (\xi )=0\) for some \(\xi <0\), then by (3.15), \(\tilde{F}^{\prime \prime }_\infty (\xi )>0\). Hence, \(x =\xi \) is a strict local minimum. This is a contradiction since \(\tilde{F}_\infty \) is increasing. Hence \(\tilde{F}^{\prime }_\infty (x) > 0\), for all \(x<0\) and as a consequence, \(\tilde{F}_\infty \) is strictly increasing on \((-\infty ,0]\).
Our next step is to prove that \(\lim \limits _{x \rightarrow -\infty } \tilde{F}_\infty (x) =0\). We consider the process \(Z\) defined by \(Z(t)=Y(t)+ \beta \), for all \(t \ge 0\), where \(Y\) is given in (3.17). Then \(Z\) is an Ornstein-Uhlenbeck process that satisfies
For each \(y>x+\beta \), we introduce the stopping time \(\tilde{\tau }_y\) by
Then \(\tau _0\), defined in (5.2), is identical to \(\tilde{\tau }_\beta \), and \(\tilde{F}_\infty (x)= E[\mathrm{{e}}^{- \gamma \tilde{\tau }_\beta } | Z(0)=x + \beta ]\), for all \(x<\min \{ - \beta , 0 \}\). Using the strong Markov property, we obtain
and
Therefore, to reach the desired conclusion, it suffices to demonstrate that
For this, consider the process \(Z\) that is characterized by \(Z(t)=y+\sigma W(t) - \mu \int _{0}^{t}Z(s)\mathrm{{d}}s\). Then it is well known that, via a random time change, one can write
where \(B\) is another Brownian motion. Next, we introduce a collection of stopping times \((T_y)\) with respect to this Brownian motion. Specifically, for each \(y<0\), let
This enables one to derive a relationship between \(\tilde{\tau }_0\) and \(T_y\), namely
The distribution of the Brownian stopping time \(T_y\) is well known (see [3]), and it will help us compute the limit \(\lim \limits _{ y \rightarrow -\infty }E[\mathrm{{e}}^{- \gamma \tilde{\tau }_0}|Z(0)=y]\).
Observe that \(\mathrm{{e}}^{- \gamma \tilde{\tau }_0}=[(\frac{2 \mu }{\sigma ^2})T_y +1]^{ - \frac{\gamma }{2 \mu }}\). It follows that
Using the bounded convergence theorem, we notice that the limit on the right-hand side vanishes since \(\lim \limits _{ y \rightarrow -\infty }T_y=\infty \) almost surely.
Consequently, \(\lim \limits _{x \rightarrow -\infty } \tilde{F}_\infty (x) =0\). Now it is clear that \(\tilde{F}_\infty \) satisfies (3.15) and (3.16). The uniqueness of solutions to (3.15) and (3.16) can be established by the fact that the difference of two solutions to (3.15) cannot have any positive local maxima. Therefore, \(\tilde{F}_\infty \) is identical to \(F_\infty \), as characterized via the initial value problem (3.15) and (3.16). It also has the stochastic representation (5.3).
In our next step, we show that \(\lim \limits _{x \rightarrow -\infty } F^{\prime }_\infty (x) =0\) and the function \(F_\infty \) is strictly convex. First we extend the function \(F_\infty \) to \((-\infty ,\infty )\), so that it satisfies the differential equation (3.15) everywhere on \((-\infty ,\infty )\). Since \(F_\infty \) is strictly increasing on \((- \infty , 0]\), by (3.16) it is clear that \(\liminf \limits _{x \rightarrow -\infty } F^{\prime }_\infty (x) =0\). Thus, we can choose a sequence \((y_n)\) strictly decreasing to \(-\infty \), such that \(y_{n+1}<y_n<0\) and \(0<F^{\prime }_\infty (y_{n+1})< F^{\prime }_\infty (y_{n})\), for all \(n\). Consequently, there is a point \(\xi _n\) such that \(y_{n+1}<\xi _n<y_n\) and \(F^{\prime \prime }_\infty (\xi _n)>0\). Note that the sequence \((\xi _n)\) is also strictly decreasing to \(-\infty \).
Let \(z=\inf \{ x \ge 0:F^{\prime }_\infty (x)\le 0 \}\). If \(z\) is finite, then \(F^{\prime }_\infty (z)=0\) and \(F^{\prime }_\infty (x)>0\), for all \(x <z\), and consequently, \(F_\infty (z) \ge F_\infty (0)=1\). By (3.15), \(Y_n\) has paths of bounded variation \(F^{\prime \prime }_\infty (z)>0\), and hence \(F_\infty \) has a strict local minimum at the point \(x=z\), which is a contradiction. Therefore, \(z\) cannot be finite and \(F^{\prime }_\infty (x)>0\), for all \(x\) in \((-\infty ,\infty )\).
Next we consider any point \(x_1>0\), such that \(x_1+\beta >0\). Then \(F_\infty (x_1)>F_\infty (0)=1\) and \(F^{\prime }_\infty (x_1) >0\). Using (3.15), we also obtain \(F^{\prime \prime }_\infty (x_1) >0\). Now introduce the function \(H(x)=F^{\prime \prime }_\infty (x)\), on the interval \([\xi _n, x_1]\). Then, by differentiating (3.15), we see that
\(H(\xi _n)>0\) and \(H(x_1)>0\). Let \(\xi _n \le c \le x_1\) so that \(H(c)=\min \limits _{[\xi _n, x_1]}H(x)\). Suppose that \(H(c)\le 0\), then \(\xi _n <c<x_1\) and \(H^{\prime }(c)=0\). If \(H(c)=0\), then by the uniqueness of the solution to the above differential equation, it follows that \(H\) is identically zero. If \(H(c)<0\), again by the above differential equation, we have \(H^{\prime \prime }(c)<0\) and \(x=c\) is a strict local maximum; this is a contradiction. Hence \(H(c)>0\) and, consequently, \(F^{\prime \prime }_\infty (x)>0\) on the interval \([\xi _n, x_1]\). But one can choose \(x_1\) arbitrarily large and the sequence \((\xi _n)\) is decreasing to \(-\infty \). We thus conclude that \(F^{\prime \prime }_\infty (x)>0\), on the interval \((-\infty , \infty )\). This, together with the fact that \(\liminf \limits _{x \rightarrow -\infty } F^{\prime }_\infty (x) =0\), implies \(\lim \limits _{x \rightarrow -\infty } F^{\prime }_\infty (x) =0\). Hence, \(F_\infty \) is a strictly convex function that satisfies all the conclusions of Lemma 3.2. This completes the proof.
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Weerasinghe, A., Mandelbaum, A. Abandonment versus blocking in many-server queues: asymptotic optimality in the QED regime. Queueing Syst 75, 279–337 (2013). https://doi.org/10.1007/s11134-013-9367-5
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DOI: https://doi.org/10.1007/s11134-013-9367-5
Keywords
- Many-server queues
- Call centers
- Halfin–Whitt (QED) heavy-traffic regime
- Local-time process
- Diffusion processes and approximations
- Asymptotic optimality.