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In the next chapter we develop fault-tolerant algorithms for the concurrent-push architecture so that even server failures can be sustained without disruption to ongoing video streams.
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Appendices A.1 Proof of Theorem 10.1
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Assume server zero starts the rst service round at time t0 . Since the server clocks are not precisely synchronized, we let di be the clock difference between server i and server zero. Hence d0 =0 and max{|di d j | | i, j} = and server i will start service round j at time (t0 + di + jT F ). Let tnew be the time a new-session request arrive at the servers. Then the request will arrive at server i during round vi : vi = tnew (t0 + di ) TF (10.45)
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and the rst video block will be retrieved at round (vi + 1) and transmitted at round (vi + 2). Hence the transmission jitter between server i and server k for stripe j can be expressed as: i,k, j = (t0 + di + (vi + 2 + j)TF ) (t0 + dk + (vk + 2 + j)TF ) Substituting equation (10.45) into (10.46) we have i,k, j = di + tnew (t0 + di ) TF TF dk + tnew (t0 + dk ) TF TF
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tnew (t0 +di ) . TF
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(10.46)
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(10.47)
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Without loss of generality, we can assume di dk and let H = i,k, j = (di + H TF ) dk + H + = (di dk ) + TF Noting that x + y x + y we have i,k, j (di dk ) + TF = (di dk ) TF H H (di dk ) TF H H+
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Then we have (10.48)
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(di dk ) TF TF (di dk ) TF
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(di dk ) TF
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(10.49)
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Finally, making use of the result that x/y y (x y) we can then obtain i,k, j = (di dk ) TF (di dk ) TF (10.50)
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(di dk ) ((di dk ) TF ) = TF and the result follows.
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A Concurrent Push-Parallel Server Architecture
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A.2 Proof of Theorem 10.2
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Let the new session request arrive at the admission scheduler at time t during group vnew = t G/TF . Then due to clock jitter, the current group at other servers can range from (t )G/TF to (t + )G/TF . To guarantee that the assigned group has not started in any of the servers implies assigning a group larger than the largest current group in any of the servers, i.e. snew = (t + )G/TF + 1. Applying the inequality x + y x + y we have snew t G/TF + G/TF + 1 Substituting into vnew we have snew vnew + G/TF + 1 and the results follows. (10.51)
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References
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[1] T.C. Chiueh, C. Venkatramani, and M. Vernick, Design and Implementation of the Stony Brook Video Server, Technical Report TR-16, Computer Science Department, State University of New York at Stony Brook, August 1995. [2] F.A. Tobagi and J. Pang, StarWorks: A Video Applications Server, IEEE COMPCON Spring 93, 1993, pp. 4 11. [3] R. Buck, The Oracle Media Server for nCube Massively Parallel Systems, Proceedings of 8th International Parallel Processing Symposium, 1994, pp. 670 673. [4] H. Taylor, D. Chin, and S. Knight, The Magic Video-on-Demand Server and Real-Time Simulation System, IEEE Parallel and Distributed Technology: Systems and Applications, vol. 3, no. 2, 1995, pp. 40 51. [5] D.J. Gemmell, H.M. Vin, D.D. Kandlur, P.V. Rangan, and L.A. Rowe, Multimedia Storage Servers: A Tutorial, IEEE Computer, vol. 28, no. 5, May 1995, pp. 40 49. [6] D. Mills, Internet Time Synchronization: The Network Time Protocol, IEEE Transactions on Comm., vol. 39, no. 10, Oct. 1991, pp. 1482 1493. [7] Z. Yang and T.A. Marsland, (eds) Global States and Time in Distributed Systems, IEEE Computer Society Press, Los Alamitos, CA, USA, 1994. [8] P.S. Yu, M.S. Chen, and D.D. Kandlur, Grouped Sweeping Scheduling for DASD-based Multimedia Storage Management, ACM Multimedia Systems, vol. 1, 1993, pp. 99 109. [9] J.N. Lloyd and K.S. Samuel, Urn Models and their Application, John Wiley & Sons, 1997, pp. 125 126.
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