Skip to main content Accessibility help
Home
Hostname: page-component-747cfc64b6-bv7lh Total loading time: 0.291 Render date: 2021-06-16T12:21:02.099Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true } Journal of Applied Probability

# Approximation of excessive backlog probabilities of two tandem queues

Published online by Cambridge University Press:  16 November 2018

## Abstract

Let X be the constrained random walk on ℤ+2 having increments (1,0), (-1,1), and (0,-1) with respective probabilities λ, µ1, and µ2 representing the lengths of two tandem queues. We assume that X is stable and µ1≠µ2. Let τn be the first time when the sum of the components of X equals n. Let Y be the constrained random walk on ℤ×ℤ+ having increments (-1,0), (1,1), and (0,-1) with probabilities λ, µ1, and µ2. Let τ be the first time that the components of Y are equal to each other. We prove that Pn-xn(1),xn(2)(τ<∞) approximates pn(xn) with relative error exponentially decaying in n for xn=⌊nx⌋, x ∈ℝ+2, 0<x(1)+x(2)<1, x(1)>0. An affine transformation moving the origin to the point (n,0) and letting n→∞ connect the X and Y processes. We use a linear combination of basis functions constructed from single and conjugate points on a characteristic surface associated with X to derive a simple expression for ℙy(τ<∞) in terms of the utilization rates of the nodes. The proof that the relative error decays exponentially in n uses a sequence of subsolutions of a related Hamilton‒Jacobi‒Bellman equation on a manifold consisting of three copies of ℝ+2 glued to each other along the constraining boundaries. We indicate how the ideas of the paper can be generalized to more general processes and other exit boundaries.

## MSC classification

Type
Research Papers
Information
Journal of Applied Probability , September 2018 , pp. 968 - 997
Copyright
Copyright © Applied Probability Trust 2018

## Access options

Get access to the full version of this content by using one of the access options below.

## References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
Atar, R. and Dupuis, P. (1999). Large deviations and queueing networks: methods for rate function identification. Stoch. Process. Appl. 84, 255296.CrossRefGoogle Scholar
Blanchet, J. (2013). Optimal sampling of overflow paths in Jackson networks. Math. Operat. Res. 38, 698719.CrossRefGoogle Scholar
Blanchet, J. H., Leder, K. and Glynn, P. W. (2008). Efficient simulation of light-tailed sums: an old-folk song sung to a faster new tune. In Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer, Berlin, pp. 227258.Google Scholar
Borovkov, A. A. and Mogul'skiĭ, A. A. (2001). Large deviations for Markov chains in the positive quadrant. Russian Math. Surveys 56, 803916.CrossRefGoogle Scholar
Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Springer, New York.CrossRefGoogle Scholar
Comets, F., Delarue, F. and Schott, R. (2007). Distributed algorithms in an ergodic Markovian environment. Random Structures Algorithms 30, 131167.CrossRefGoogle Scholar
Dai, J. G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Systems 1, 146208.CrossRefGoogle Scholar
De Boer, P.-T. (2006). Analysis of state-independent importance-sampling measures for the two-node tandem queue. ACM Trans. Model. Comput. Simul. 16, 225250.CrossRefGoogle Scholar
Dean, T. and Dupuis, P. (2009). Splitting for rare event simulation: a large deviation approach to design and analysis. Stoch. Process. Appl. 119, 562587.CrossRefGoogle Scholar
Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.CrossRefGoogle Scholar
Dupuis, P. and Ellis, R. S. (1995). The large deviation principle for a general class of queueing systems. I. Trans. Amer. Math. Soc. 347, 26892751.Google Scholar
Dupuis, P. and Wang, H. (2004). Importance sampling, large deviations, and differential games. Stoch. Stoch. Reports 76, 481508.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Operat. Res. 32, 723757.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2009). Importance sampling for Jackson networks. Queueing Systems 62, 113157.CrossRefGoogle Scholar
Dupuis, P., Leder, K. and Wang, H. (2007). Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17, 14.CrossRefGoogle Scholar
Dupuis, P., Sezer, A. D. and Wang, H. (2007). Dynamic importance sampling for queueing networks. Ann. Appl. Prob. 17, 13061346.CrossRefGoogle Scholar
Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury, Belmont, CA.Google Scholar
Flajolet, P. (1986). The evolution of two stacks in bounded space and random walks in a triangle. In Mathematical Foundations of Computer Science, 1986, Springer, Berlin, pp. 325340.CrossRefGoogle Scholar
Foley, R. D. and McDonald, D. R. (2012). Constructing a harmonic function for an irreducible nonnegative matrix with convergence parameter R > 1. Bull. London Math. Soc. 44, 533544.CrossRefGoogle Scholar
Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 2nd edn. Springer, Heidelberg.CrossRefGoogle Scholar
Glasserman, P. and Kou, S.-G. (1995). Analysis of an importance sampling estimator for tandem queues. ACM Trans. Model. Comput. Simul. 5, 2242.CrossRefGoogle Scholar
Griffiths, P. A. (1989). Introduction to Algebraic Curves. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Guillotin-Plantard, N. and Schott, R. (2006). Dynamic Random Walks: Theory and Applications. Elsevier, Amsterdam.CrossRefGoogle Scholar
Henderson, S. G. and Nelson, B. L. (eds) (2006). Handbooks in Operations Research and Management Science: Vol. 13, Simulation. North-Holland, Amsterdam.Google Scholar
Ignatiouk-Robert, I. (2000). Large deviations of Jackson networks. Ann. Appl. Prob. 10, 9621001.Google Scholar
Ignatiouk-Robert, I. and Loree, C. (2010). Martin boundary of a killed random walk on a quadrant. Ann. Prob. 38, 11061142.CrossRefGoogle Scholar
Ignatyuk, I. A., Malyshev, V. A. and Scherbakov, V. V. (1994). Boundary effects in large deviation problems. Russian. Math. Surveys 49, 4199.CrossRefGoogle Scholar
Juneja, S. and Nicola, V. (2005). Efficient simulation of buffer overflow probabilities in Jackson networks with feedback. ACM Trans. Model. Comput. Simul. 15, 281315.CrossRefGoogle Scholar
Knuth, D. E. (1969). The Art of Computer Programming, Vol. 1, Fundamental Algorithms. Addison-Wesley, Reading, MA.Google Scholar
Kobayashi, M. and Miyazawa, M. (2013). Revisiting the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. In Matrix-Analytic Methods in Stochastic Models, Springer, New York, pp. 145185.CrossRefGoogle Scholar
Kurkova, I. A. and Malyshev, V. A. (1998). Martin boundary and elliptic curves. Markov Process. Relat. Fields 4, 203272.Google Scholar
Kushner, H. J. and Dupuis, P. (2001). Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Louchard, G. and Schott, R. (1991). Probabilistic analysis of some distributed algorithms. Random Structures Algorithms 2, 151186.CrossRefGoogle Scholar
Louchard, G., Schott, R., Tolley, M. and Zimmermann, P. (1994). Random walks, heat equation and distributed algorithms. J. Comput. Appl. Math. 53, 243274.CrossRefGoogle Scholar
Maier, R. S. (1991). Colliding stacks: a large deviations analysis. Random Structures Algorithms 2, 379420.CrossRefGoogle Scholar
Maier, R. S. (1993). Large fluctuations in stochastically perturbed nonlinear systems: applications in computing. In 1992 Lectures in Complex Systems, Addison-Wesley, Reading, MA, pp. 501517.Google Scholar
McDonald, D. R. (1999). Asymptotics of first passage times for random walk in an orthant. Ann. Appl. Prob. 9, 110145.Google Scholar
Miretskiy, D., Scheinhardt, W. and Mandjes, M. (2010). State-dependent importance sampling for a Jackson tandem network. ACM Trans. Model. Comput. Simul. 20, 15.CrossRefGoogle Scholar
Miyazawa, M. (2009). Tail decay rates in double QBD processes and related reflected random walks. Math. Operat. Res. 34, 547575.CrossRefGoogle Scholar
Miyazawa, M. (2011). Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP 19, 233299.CrossRefGoogle Scholar
Ney, P. and Nummelin, E. (1987). Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Prob. 15, 561592.CrossRefGoogle Scholar
Nicola, V. F. and Zaburnenko, T. S. (2007). Efficient importance sampling heuristics for the simulation of population overflow in Jackson networks. ACM Trans. Model. Comput. Simul. 17, 10.CrossRefGoogle Scholar
Parekh, S. and Walrand, J. (1989). A quick simulation method for excessive backlogs in networks of queues. IEEE Trans. Automatic Control 34, 5466.CrossRefGoogle Scholar
Revuz, D. (1984). Markov Chains, 2nd edn. North-Holland, Amsterdam.Google Scholar
Ridder, A. (2009). Importance sampling algorithms for first passage time probabilities in the infinite server queue. Europ. J. Operat. Res. 199, 176186.CrossRefGoogle Scholar
Robert, P. (2003). Stochastic Networks and Queues. Springer, Berlin.CrossRefGoogle Scholar
Rubino, G. and Tuffin, B. (2009). Rare Event Simulation using Monte Carlo Methods. John Wiley, New York.CrossRefGoogle Scholar
Setayeshgar, L. and Wang, H. (2013). Efficient importance sampling schemes for a feed-forward network. ACM Trans. Model. Comput. Simul. 23, 21.CrossRefGoogle Scholar
Sezer, A. D. (2006). Dynamic Importance Sampling for Queueing Networks. Doctoral thesis. Division of Applied Mathematics, Brown University.Google Scholar
Sezer, A. D. (2007). Asymptotically optimal importance sampling for Jackson networks with a tree topology. Preprint. Available at https://arxiv.org/abs/0708.3260.Google Scholar
Sezer, A. D. (2009). Importance sampling for a Markov modulated queuing network. Stoch. Process. Appl. 119, 491517.CrossRefGoogle Scholar
Sezer, A. D. (2010). Asymptotically optimal importance sampling for Jackson networks with a tree topology. Queueing Systems 64, 103117.CrossRefGoogle Scholar
Sezer, A. D. (2015). Exit probabilities and balayage of constrained random walks. Preprint. Available at https://arxiv.org/abs/1506.08674.Google Scholar
Sezer, A. D. and Özbudak, F. (2011). Approximation of bounds on mixed-level orthogonal arrays. Adv. Appl. Prob. 43, 399421.CrossRefGoogle Scholar
Yao, A. C. (1981). An analysis of a memory allocation scheme for implementing stacks. SIAM J. Comput. 10, 398403.CrossRefGoogle Scholar
2
Cited by

# Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Approximation of excessive backlog probabilities of two tandem queues
Available formats
×

# Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Approximation of excessive backlog probabilities of two tandem queues
Available formats
×

# Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Approximation of excessive backlog probabilities of two tandem queues
Available formats
×
×

#### Reply to:Submit a response

Please enter your response.

#### Your details

Please enter a valid email address.

#### Conflicting interests

Do you have any conflicting interests? *