Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

We consider a Lévy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Lévy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275–298, 2007 and Queueing Syst. 60:203–226, 2008) for the corresponding tandem queue without an intermediate input.

[1]  Florin Avram,et al.  Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results. , 2008, 0802.4060.

[2]  Lokenath Debnath,et al.  Introduction to the Theory and Application of the Laplace Transformation , 1974, IEEE Transactions on Systems, Man, and Cybernetics.

[3]  P. Protter.  Stochastic integration and differential equations , 1990 .

[4]  Ward Whitt,et al.  Asymptotics for M/G/1 low-priority waiting-time tail probabilities , 1997, Queueing Syst. Theory Appl..

[5]  Michel Mandjes,et al.  Asymptotic analysis of Lévy-driven tandem queues , 2008, Queueing Syst. Theory Appl..

[6]  J. Harrison,et al.  Reflected Brownian Motion in an Orthant: Numerical Methods for Steady-State Analysis , 1992 .

[7]  Cheng-Shang Chang,et al.  Sample path large deviations and intree networks , 1995, Queueing Syst. Theory Appl..

[8]  P. Lieshout,et al.  Tandem Brownian queues , 2007, Math. Methods Oper. Res..

[9]  W. Burnside.  Theory of Functions , 1899, Nature.

[10]  Steven P. Lalley.  Return probabilities for random walk on a half-line , 1995 .

[11]  W. Sutton.  The Asymptotic Expansion of a Function Whose Operational Equivalent is Known , 1934 .

[12]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[13]  Krzysztof Debicki,et al.  Quasi-Product Forms for Lévy-Driven Fluid Networks , 2007, Math. Oper. Res..

[14]  Kurt Majewski.  Large deviations of the steady-state distribution of reflected processes with applications to queueing systems , 1998, Queueing Syst. Theory Appl..

[15]  Philip Heidelberger,et al.  Effective Bandwidth and Fast Simulation of ATM Intree Networks , 1994, Perform. Evaluation.

[16]  Ruth J. Williams,et al.  Brownian Models of Open Queueing Networks with Homogeneous Customer Populations , 1987 .

[17]  Александр Алексеевич Боровков,et al.  Большие уклонения для цепей Маркова в положительном квадранте@@@Large deviations for Markov chains in the positive quadrant , 2001 .

[18]  Ward Whitt,et al.  An Introduction to Stochastic-Process Limits and their Application to Queues , 2002 .

[19]  W. Whitt,et al.  Useful martingales for stochastic storage processes with Lévy input , 1992, Journal of Applied Probability.

[20]  K. Knopp.  Theory of Functions , 1958 .

[21]  Florin Avram,et al.  Explicit Solutions for Variational Problems in the Quadrant , 2001, Queueing Syst. Theory Appl..

[22]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[23]  David D. Yao,et al.  Fundamentals of Queueing Networks , 2001 .

[24]  Masakiyo Miyazawa,et al.  Rate conservation laws: A survey , 1994, Queueing Syst. Theory Appl..

[25]  Masakiyo Miyazawa,et al.  Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks , 2009, Math. Oper. Res..

[26]  Onno Boxma,et al.  Useful Martingales for Stochastic Storage Processes with Lévy-Type Input , 2013, Journal of Applied Probability.

[27]  Jordan Stoyanov,et al.  Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization , 2003 .