Scheduling reserved traffic in input-queued switches: new delay bounds via probabilistic techniques

We consider the problem of providing delay bounds to reserved traffic in high-speed input-queued switches. We assume that the matrix of bandwidth demands is known and we use the now standard approach of decomposing this matrix into a convex combination of permutation matrices. Our problem therefore reduces to the problem of constructing a schedule for these permutation matrices. In this paper we derive delay bounds for four algorithms that are based on probabilistic techniques. For each algorithm we first place tokens randomly in continuous time for each permutation matrix. If the nth token that appears corresponds to permutation matrix M/sub k/ then we schedule matrix M/sub k/ in the nth time slot. The algorithms differ in how the random token processes are defined. For two of the algorithms we are able to perform a derandomization so as to obtain deterministic schedules. We show through numerical computation that in many situations the resulting delay bounds are smaller than the previously best-known delay bounds of Chang, Chen, and Huang (1999).

[1]  Leandros Tassiulas,et al.  QoS provisioning and tracking fluid policies in input queueing switches , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[2]  F. Knight.  Essentials of Brownian Motion and Diffusion , 1981 .

[3]  Jean-Yves Le Boudec,et al.  Network Calculus , 2001, Lecture Notes in Computer Science.

[4]  Jean-Yves Le Boudec,et al.  Delay jitter bounds and packet scale rate guarantee for expedited forwarding , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[5]  Cheng-Shang Chang,et al.  Birkhoff-von Neumann input buffered crossbar switches , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[6]  Balaji Prabhakar,et al.  Matching Output Queueing with Combined Input and Output Queueing , 1999 .

[7]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[8]  Jean C. Walrand,et al.  Achieving 100% throughput in an input-queued switch , 1996, Proceedings of IEEE INFOCOM '96. Conference on Computer Communications.

[9]  P. Sen,et al.  Theory of rank tests , 1969 .

[10]  Marco Ajmone Marsan,et al.  Bounds on average delays and queue size averages and variances in input-queued cell-based switches , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[11]  James Ryan Giles.  Scheduling Multirate Periodic Traffic in a Packet Switch , 1997 .

[12]  Hideaki Takagi,et al.  Analysis and Application of Polling Models , 2000, Performance Evaluation.

[13]  J. Doob.  Heuristic Approach to the Kolmogorov-Smirnov Theorems , 1949 .

[14]  Abhay Parekh,et al.  A generalized processor sharing approach to flow control in integrated services networks-the single node case , 1992, [Proceedings] IEEE INFOCOM '92: The Conference on Computer Communications.

[15]  Cheng-Shang Chang,et al.  On service guarantees for input-buffered crossbar switches: a capacity decomposition approach by Birkhoff and von Neumann , 1999, 1999 Seventh International Workshop on Quality of Service. IWQoS'99. (Cat. No.98EX354).

[16]  Thomas L. Saaty,et al.  Elements of queueing theory , 2003 .

[17]  W. Vervaat,et al.  A Relation between Brownian Bridge and Brownian Excursion , 1979 .

[18]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[19]  Russ Bubley,et al.  Randomized algorithms , 1995, CSUR.

[20]  J. W. Roberts,et al.  Performance evaluation and design of multiservice networks , 1992 .

[21]  John von Neumann,et al.  1. A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem , 1953 .

[22]  A. Charny,et al.  Algorithms for providing bandwidth and delay guarantees in input-buffered crossbars with speedup , 1998, 1998 Sixth International Workshop on Quality of Service (IWQoS'98) (Cat. No.98EX136).

[23]  Hui Zhang,et al.  Exact emulation of an output queueing switch by a combined input output queueing switch , 1998, 1998 Sixth International Workshop on Quality of Service (IWQoS'98) (Cat. No.98EX136).

[24]  Nirwan Ansari,et al.  Input-queued switching with QoS guarantees , 1999, IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. The Future is Now (Cat. No.99CH36320).

[25]  Mary C. Brennan,et al.  on the , 1982 .

[26]  Annie Gravey,et al.  On the Geo/D/1/and Geo/D/1/n Queues , 1990, Perform. Evaluation.

[27]  Balaji Prabhakar,et al.  The throughput of data switches with and without speedup , 2000, Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No.00CH37064).

[28]  Nick McKeown,et al.  ATM input-buffered switches with the guaranteed-rate property , 1998, Proceedings Third IEEE Symposium on Computers and Communications. ISCC'98. (Cat. No.98EX166).

[29]  John Odentrantz,et al.  Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues , 2000, Technometrics.

[30]  Van Jacobson,et al.  An Expedited Forwarding PHB , 1999, RFC.

[31]  Leandros Tassiulas,et al.  Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks , 1992 .

[32]  Devavrat Shah,et al.  Delay bounds for approximate maximum weight matching algorithms for input queued switches , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[33]  Bruce Hajek.  A Queue with Periodic Arrivals and Constant Service Rate , 1994 .

[34]  Cheuk H. Lam,et al.  Path Switching - A Quasi-Static Routing Scheme for Large-Scale ATM Packet Switches , 1997, IEEE J. Sel. Areas Commun..