We consider a pair of coupled queues driven by independent spectrally-positive Lévy processes. With respect to the bi-variate workload process this framework includes both the coupled processor model and the two-server fluid network with independent Lévy inputs. We identify the joint transform of the stationary workload distribution in terms of Wiener-Hopf factors corresponding to two auxiliary Lévy processes with explicit Laplace exponents. We reinterpret and extend the ideas of Cohen and Boxma (1983) to provide a general and uniform result with a neat transform expression.
References
Blanc, J. P. C. (1987). A numerical study of a coupled processor model. In Computer Performance and Reliability ( G. Iazeolla, P.-J. Courtois and O. J. Boxma, eds.) 289–303.Blanc, J. P. C. (1987). A numerical study of a coupled processor model. In Computer Performance and Reliability ( G. Iazeolla, P.-J. Courtois and O. J. Boxma, eds.) 289–303.
Borst, S. C., Boxma, O. J. and Jelenkovic, P. R. (2003). Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows. Queueing Systems 43 273–306. MR1976261 10.1023/A:1023237129453Borst, S. C., Boxma, O. J. and Jelenkovic, P. R. (2003). Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows. Queueing Systems 43 273–306. MR1976261 10.1023/A:1023237129453
Cohen, J. W. and Boxma, O. J. (1983). Boundary Value Problems in Queueing System Analysis. North-Holland Mathematics Studies 79. North-Holland Publishing Co., Amsterdam. MR703000Cohen, J. W. and Boxma, O. J. (1983). Boundary Value Problems in Queueing System Analysis. North-Holland Mathematics Studies 79. North-Holland Publishing Co., Amsterdam. MR703000
Debicki, K., Dieker, A. B. and Rolski, T. (2007). Quasi-product forms for Lévy-driven fluid networks. Math. Oper. Res. 32 629–647. MR2348239 10.1287/moor.1070.0259Debicki, K., Dieker, A. B. and Rolski, T. (2007). Quasi-product forms for Lévy-driven fluid networks. Math. Oper. Res. 32 629–647. MR2348239 10.1287/moor.1070.0259
Den Iseger, P., Gruntjes, P. and Mandjes, M. (2013). A Wiener-Hopf based approach to numerical computations in fluctuation theory for Lévy processes. Math. Meth. Oper. Res. 33 1–18. MR3073952 10.1007/s00186-013-0434-9Den Iseger, P., Gruntjes, P. and Mandjes, M. (2013). A Wiener-Hopf based approach to numerical computations in fluctuation theory for Lévy processes. Math. Meth. Oper. Res. 33 1–18. MR3073952 10.1007/s00186-013-0434-9
Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47 325–351. MR525314Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47 325–351. MR525314
Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308. MR606992 10.1214/aop/1176994471 euclid.aop/1176994471
Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308. MR606992 10.1214/aop/1176994471 euclid.aop/1176994471
Hooghiemstra, G., Keane, M. and van de Ree, S. (1988). Power series for stationary distributions of coupled processor models. SIAM J. Appl. Math. 48 1159–1166. MR960477 10.1137/0148069Hooghiemstra, G., Keane, M. and van de Ree, S. (1988). Power series for stationary distributions of coupled processor models. SIAM J. Appl. Math. 48 1159–1166. MR960477 10.1137/0148069
Kella, O. (1996). Stability and nonproduct form of stochastic fluid networks with Lévy inputs. Ann. Appl. Probab. 6 186–199. MR1389836 10.1214/aoap/1034968070 euclid.aoap/1034968070
Kella, O. (1996). Stability and nonproduct form of stochastic fluid networks with Lévy inputs. Ann. Appl. Probab. 6 186–199. MR1389836 10.1214/aoap/1034968070 euclid.aoap/1034968070
Kella, O. and Ramasubramanian, S. (2012). Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant. Math. Oper. Res. 37 301–312. MR2931282 10.1287/moor.1120.0538Kella, O. and Ramasubramanian, S. (2012). Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant. Math. Oper. Res. 37 301–312. MR2931282 10.1287/moor.1120.0538
Kella, O. and Whitt, W. (1992a). A tandem fluid network with Lévy input. In Queueing and Related Models, ( U. N. Bhat and I. V. Basawa, eds.) 9 112–128. MR1210563Kella, O. and Whitt, W. (1992a). A tandem fluid network with Lévy input. In Queueing and Related Models, ( U. N. Bhat and I. V. Basawa, eds.) 9 112–128. MR1210563
Kella, O. and Whitt, W. (1992b). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probab. 29 396–403. MR1165224 10.2307/3214576Kella, O. and Whitt, W. (1992b). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probab. 29 396–403. MR1165224 10.2307/3214576
Konheim, A. G., Meilijson, I. and Melkman, A. (1981). Processor-sharing of two parallel lines. J. Appl. Probab. 18 952–956. MR633243 10.2307/3213071Konheim, A. G., Meilijson, I. and Melkman, A. (1981). Processor-sharing of two parallel lines. J. Appl. Probab. 18 952–956. MR633243 10.2307/3213071
Miyazawa, M. and Rolski, T. (2009). Tail asymptotics for a Lévy-driven tandem queue with an intermediate input. Queueing Syst. 63 323–353. MR2576017 10.1007/s11134-009-9146-5Miyazawa, M. and Rolski, T. (2009). Tail asymptotics for a Lévy-driven tandem queue with an intermediate input. Queueing Syst. 63 323–353. MR2576017 10.1007/s11134-009-9146-5
