Spectral Gap of the Erlang A Model in the Halfin-Whitt Regime

References

• M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions (10th printing), 1972.Google Scholar
• J. D. Atkinson and T. K. Caughley. Spectral density of piecewise linear first order systems excited by white noise. Int. J. Non-Linear Mechanics, 3:137–156, 1968. MR0230391Google Scholar
• J. D. Atkinson. Spectral density of first order piecewise linear systems excited by white noise. PhD thesis, CalTech, 1967.Google Scholar
• J. P. C. Blanc and E. A. van Doorn. Relaxation times for queueing systems. In Mathematics and Computer Science (eds. J.W. de Bakker, M. Hazewinkel, J.K. Lenstra), North-Holland, Amsterdam, 139–162, 1984. MR0873577Google Scholar
• N. Bleistein and R. A. Handelsman. Asymptotic Expansions of Integrals. Dover, New York, 1986. MR0863284Google Scholar
• E. A. Coddington and N. Levinson. Theory of Ordinary Differential Equations. McGraw–Hill, New York, 1955. MR0069338Google Scholar
• J. W. Cohen. The Single Server Queue. North Holland, Amsterdam, 1982. MR0668697Google Scholar
• J. G. Dai, S. He and T. Tezcan. Many-server diffusion limits for G/Ph/n + M queues. Preprint, 2008.Google Scholar
• E. A. van Doorn. Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Advances in Applied Probability 17:514–530, 1985. MR0798874Google Scholar
• A. Erdelyi. Higher Transcendental Functions. Vol. 2. MacGraw-Hill, New York, 1953.Google Scholar
• P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, Cambridge, 2009. MR2483235Google Scholar
• C. Fricker, Ph. Robert and D. Tibi. On the rates of convergence of Erlang’s model. Journal of Applied Probability 36:1167–1184, 1999. MR1742158Google Scholar
• N. Gans, G. Koole and A. Mandelbaum. Telephone call centers: Tutorial, review and research prospects. Manufacturing and Service Operations Management 5:79–141, 2003.Link, Google Scholar
• D. Gamarnik and D. A. Goldberg. On the exponential rate of convergence to stationarity in the Halfin-Whitt regime I: The spectral gap of the M/M/n queue. Preprint, 2008.Google Scholar
• O. Garnett, A. Mandelbaum and M. Reiman. Designing a call center with impatient customers. Manufacturing & Service Operations Management 4:208–227, 2002.Link, Google Scholar
• I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. 5th ed., Academic Press, New York, 1994.Google Scholar
• S. Halfin and W. Whitt. Heavy-traffic limits for queues with many exponential servers. Operations Research 29:567–588, 1981. MR0629195Link, Google Scholar
• D. Iglehart. Limiting diffusion approximations for the many server queue and the repairman problem. Journal of Applied Probability 2:429–441, 1965. MR0184302Google Scholar
• W. Kang and K. Ramanan. Fluid limits of many-server queues with reneging. Ann. Appl. Prob. 20:2204–2260, 2010. MR2759733Google Scholar
• S. Karlin and J. L. McGregor. Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8:87–118, 1958. MR0097132Google Scholar
• J. Keilson. A review of transient behavior in regular diffusion and birth-death processes. J. Applied. Prob. 1:247–266, 1964. MR0181008Google Scholar
• J. Lehmann, P. Reimann, and P. Hänggi. Surmounting oscillating barriers: Path integral approach for weak noise. Phys. Rev. E, 62:6282–6303, 2000.Google Scholar
• J. S. H. van Leeuwaarden and C. Knessl. Transient analysis of the Halfin-Whitt diffusion. Stochastic Processes and their Applications 121: 1524–1545, 2011. MR2802464Google Scholar
• V. Linetsky. On the transition densities for reflected diffusions. Adv. Appl. Prob. 37:435–460, 2005. MR2144561Google Scholar
• C. Maglaras and A. Zeevi. Diffusion approximations for a multiclass Markovian service system with “guaranteed” and “best-effort” service levels. Math. Oper. Res. 29:786–813, 2004. MR2104155Link, Google Scholar
• A. N. Malakhov and A. L. Pankratov. Exact solution of Kramers’ problem for piecewise parabolic potentials. Physica A, 229:109–126, 1996.Google Scholar
• J. Reed. The G/GI/N queue in the Halfin-Whitt regime. Ann. Appl. Prob. 19: 2211–2269, 2009. MR2588244Google Scholar
• W. T. Reid. Sturmian Theory for Ordinary Differential Equations. Springer-Verlag, New York–Berlin, 1980. MR0606199Google Scholar
• I. Stakgold. Boundary Value Problems of Mathematical Physics, volume I. MacMillan, New York, 1967. MR0205776Google Scholar
• W. Szpankowski. Average Case Analysis of Algorithms on Sequences. Wiley-Interscience, New York, 2001. MR1816272Google Scholar
• N. M. Temme. Parabolic cylinder function. In R. F. Boisvert et al.., editors, NIST Handbook of Mathematical Functions. Cambridge University Press, 2010. ISBN 978-0521192255.Google Scholar
• E. C. Titchmarsh. On the discreteness of the spectrum associated with certain differential equations. Ann. Math. Pure Applied, 28:141–147, 1949. MR0036917Google Scholar
• E. C. Titchmarsh. On the discreteness of the spectrum of a differential equation. Acta Sci. Math. Szeged, 12:16–18, 1950. MR0036403Google Scholar
• E. C. Titchmarsh. Eigenfunction Expansions Associated with Second-order Differential Equations, Part I. Clarendon Press, Oxford, second edition, 1962. MR0176151Google Scholar
• A. Ward and P. Glynn. Properties of the reflected Ornstein-Uhlenbeck process. Queueing Systems 44:109–123, 2003. MR1993278Google Scholar
• W. Whitt. Efficieny-deriven heavy-traffic approximations for many-server queues with abandonments. Management Science 50:1449–1461, 2004Link, Google Scholar
• W. Whitt. Heavy-traffic limits for the G/H2∗/n/m queue. Mathematics of Operations Research 30:1–27, 2006. MR2125135Google Scholar
• W. Whitt. Fluid limits for many-server queues with abandonments. Operations Research 54:363–372, 2006. MR2127407Google Scholar
• R. Wong. Asymptotic Approximation of Integrals. Academic Press, Inc., Boston, 1989. MR1016818Google Scholar
• S. Xie and C. Knessl. On the transient behavior of the Erlang loss model: Heavy usage asymptotics. SIAM J. Appl. Math., 53:555–599, 1993. MR1212763Google Scholar
• S. Zeltyn and A. Mandelbaum. The impact of customers’ patience on delay and abandonment: some empirically-driven experiments with the M/M/n + G queue. OR Spectrum 26:377–411, 2004. MR2064063Google Scholar
• S. Zeltyn and A. Mandelbaum. Call centers with impatient customers: many-server asymptotics of the M/M/n + G queue. Queueing Systems 51:361–402, 2005. MR2189598Google Scholar