• Publication
  • 2002
  • Issue No. 4 - April
  • Abstract - Bounding of Performance Measures for Threshold-Based Queuing Systems: Theory and Application to Dynamic Resource Management in Video-on-Demand Servers
 This Article 
 Bibliographic References 
 Add to: 
Bounding of Performance Measures for Threshold-Based Queuing Systems: Theory and Application to Dynamic Resource Management in Video-on-Demand Servers
April 2002 (vol. 51 no. 4)
pp. 353-372

In this paper, we consider a K-server threshold-based queuing system with hysteresis in which the number of active servers is governed by a forward threshold vector and a reverse threshold vector. There are many applications where a threshold-based queuing system can be of great use. The main motivation for using a threshold-based approach in such applications is that they incur significant server setup, usage, and removal costs. As in most practical situations, an important concern is not only the system performance, but rather its cost/performance ratio. The motivation for use of hysteresis is to control the cost during momentary fluctuations in workload. An important and distinguishing characteristic of our work is that, in our model, we consider the time to add a server to be nonnegligible. This is a more accurate model, for many applications, than previously considered in other works. Our goal in this work is to develop an efficient method for computing the steady state probabilities of a multiserver threshold-based queuing system with hysteresis, which will, in turn, allow computation of various performance measures. We also illustrate how to apply this methodology in evaluation of the performance of a Video-on-Demand (VOD) storage server which dynamically manages its I/O resources.

[1] S. Berson, S. Ghandeharizadeh, R.R. Muntz, and X. Ju, “Staggered Striping in Multimedia Information Systems,” Proc. SIGMOD, 1994.
[2] C. Chou, L. Golubchik, and J.C.S. Lui, Striping Doesn't Scale: How to Achieve Scalability for Continuous Media Servers with Replication Proc. Int'l Conf. Distributed Computing Systems (ICDCS), pp. 64-71, Apr. 2000.
[3] P.J. Courtois, Decomposability—Queueing and Computer System Applications. New York: Academic Press, 1977.
[4] W.J. Bolosky et al., “The Tiger Video Fileserve,” Technical Report MSR-TR-96-09, Michrosoft Research, 1996.
[5] S. Ghandeharizadeh and R.R. Muntz, “Design and Implementation of Scalable Continuous Media Servers,” special issue of Parallel Computing J. Parallel Data Servers and Applications, Jan. 1998.
[6] L. Golubchik and J.C.S. Lui, “Bounding of Performance Measures for a Threshold-Based Queuing System with Hysteresis,” Proc. 1997 ACM SIGMETRICS Conf., June 1997.
[7] W. Grassman, “Transient Solutions of Markovian Queuing Systems,” Computer&Operations Research, vol. 4, pp. 47-53, 1977.
[8] S.C. Graves and J. Keilson, “The Compensation Method Applied to a One-Product Production/Inventory Problem,” J. Math. Operational Research, vol. 6, pp. 246-262, 1981.
[9] R.L. Haskin, “Tiger Shark: A Scalable File System for Multimedia,” technical report, IBM Research, 1996.
[10] O.C. Ibe, “An Approximate Analysis of a Multi-Server Queuing System with a Fixed Order of Access,” Technical Report RC9346, IBM Research, 1982.
[11] O.C. Ibe and J. Keilson, “Multi-Server Threshold Queues with Hysteresis,” Performance Evaluation, vol. 21, pp. 185-212, 1995.
[12] O.C. Ibe and K. Maruyama, “An Approximation Method for a Class of Queuing Systems,” Performance Evaluation, vol. 5, pp. 15-27, 1985.
[13] J. Keilson, Green's Function Methods in Probability Theory. London: Charles Griffin, 1965.
[14] J. Keilson, Markov Chain Models: Rarity and Exponentiality. New York: Springer, 1979.
[15] P.J.B. King, Computer and Communication Systems Performance Modelling. New York: Prentice Hall, 1990.
[16] L. Kleinrock, Queueing Systems, vol. 1.Wiley-Interscience, 1975.
[17] R.L. Larsen and A.K. Agrawala, “Control of a Heterogeneous Two-Server Exponential Queuing System,” IEEE Trans. Software Eng., vol. 9, pp. 552-526, 1983.
[18] P.W.K. Lie, J.C.S. Lui, and L. Golubchik, “Threshold-Based Dynamic Replication in Large-Scale Video-on-Demand Systems,” Proc. Eighth Int'l Workshop Research Issues in Data Eng.: Continuous-Media Databases and Applications (RIDE '98), Feb. 1998.
[19] P.W.K. Lie, J.C.S. Lui, and L. Golubchik, “Threshold-Based Dynamic Replication in Large-Scale Video-on-Demand Systems,” J. Multimedia Tools and Applications, vol. 11, no. 1, May 2000.
[20] W. Lin and P.R. Kumar,"Optimal control of a queueing system with two heterogeneous servers," IEEE Trans. Comm., vol. 29, no. 8, pp. 696-703, Aug. 1984.
[21] T. Lindvall, Lectures on the Coupling Method. Wiley Interscience, 1992.
[22] J.D.C. Little, “A Proof of the Queueing Formula$L=\lambda W$,” Operations Research, vol. 9, pp. 383-387, May 1961.
[23] J.C.S. Lui and L. Golubchik, “Stochastic Complement Analysis of Multi-Server Threshold Queues with Hysteresis,” Performance Evaluation J., vol. 35, nos. 1-2, pp. 19-48, 1999.
[24] R.R. Muntz and J.C.S. Lui, “Bounding the Response Time of a Minimum Expected Delay Routing System,” IEEE Trans. Computers, vol. 44, no. 5, pp. 1371-1382, Dec. 1995.
[25] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. Baltimore: Academic Press, 1979.
[26] C.D. Meyer, “Stochastic Complementation, Uncoupling Markov Chains and the Theory of Nearly Reducible Systems,” SIAM Rev., vol. 31, no. 2, pp. 240-272, 1989.
[27] J.A. Morrison, “Two-Server Queue with One Server Idle below a Threshold,” Queueing Systems, vol. 7, pp. 325-336, 1990.
[28] R.R. Muntz, E. de Souza e Silva, and A. Goyal, “Bounding Availability of Repairable Computer Systems,” Proc. PERFORMANCE '89 and 1989 ACM SIGMETRICS Conf., May 1989.
[29] R. Nelson and D. Towsley, “Approximating the Mean Time in System in a Multiple-Server Queue that Uses Threshold Scheduling,” Operations Research, vol. 35, pp. 419-427, 1987.
[30] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models—An Algorithmic Approach. Baltimore: John Hopkins Univ. Press, 1981.
[31] J.R. Santos and R.R. Muntz, "Performance Analysis of the RIO Multimedia Storage System with Heterogeneous Disk Configurations," Proc. 6th ACM Int'l Multimedia Conference (ACM MM 98), ACM Press, New York, 1998, pp. 303-308.
[32] W.J. Stewart, Introduction to Numerical Solution of Markov Chains. Princeton Univ. Press, 1994.
[33] D. Towsley, Application of Majorization to Control Problems in Queueing Systems, in Scheduling Theory and Its Applications, P. Chretienne, E.G. Coffman Jr., J.K. Lenstra, and Z. Liu, eds. Chichester, U.K.: Wiley, 1995.
[34] J.L. Wolf, P.S. Yu, and H. Shachnai, “DASD Dancing: A Disk Load Balancing Optimization Scheme for Video-on-Demand Computer Systems,” Proc. ACM SIGMETRICS Conf., May 1995.

Index Terms:
threshold-based systems, Markov chains, error bounds, matrix-geometric, video-on-demand systems
L. Golubchik, J.C.S. Lui, "Bounding of Performance Measures for Threshold-Based Queuing Systems: Theory and Application to Dynamic Resource Management in Video-on-Demand Servers," IEEE Transactions on Computers, vol. 51, no. 4, pp. 353-372, April 2002, doi:10.1109/12.995445
Usage of this product signifies your acceptance of the Terms of Use.