Introduction Token Bucket Regulator (TBR) used at network ingress to smoothen subscriber traffic

1. Entropy-Optimal Generalized Token Bucket Regulator Ashutosh Deepak Gore and Abhay Karandikar Department of Electrical Engineering, Indian Institute of Technology - Bombay • System Model • Conforming packet length: • Token evolution equation: • A generalized token bucket regulator is denoted by R(N, r, B), where • N = number of slots • r = (r0 , r1 , …, rN-1 ) = token increment sequence • B = (B0 , B1 , …, BN-2 ) = bucket depth sequence • Introduction • Token Bucket Regulator (TBR) used at network ingress to smoothen subscriber traffic • TBR is a regulator or a linearly bounded arrival process: • IETF (standard) TBR is defined by a token increment rate r and a bucket depth (maximum burst size) B • A generalized token bucket regulator (GTBR) is defined by a token increment sequence r and a bucket depth sequence B • used to regulate variable bit rate traffic • analogous to time-varying leaky-bucket shaper [3] • Results • Theoretical • For an optimal GTBR, equality must hold in • , except when N is small. • Computation • A generalized token bucket regulator can achieve higher information utility than a standard token bucket regulator. • The optimal bucket depth sequence B* is uniform or near-uniform. • The optimal token increment sequence r*is a decreasing sequence and is non-uniform. • Problem Statement • Can a generalized token bucket regulator achieve higher information utility than a standard IETF token bucket regulator? • Given STBR Rs (N, r, B), determine r and B of GTBR Rg (N, r, B) subject to • (token bandwidth constraint) • (burst bandwidth constraint) • (practical) • . • Variation of Information Utility with r and B • A GTBR can achieve higher information utility than an STBR because the probability mass functions (pmf’s) of the packet lengths at each stage • have a larger support • are closer to the uniform pmf • Flow Entropy Equation • Information utility is the maximum entropy achievable by any flow which is constrained by the GTBR R(N, r, B) • Entropy Hkis a function of system state uk • If a packet of length lk bits is transmitted with probability • overt information transmitted = lkbits • covert information = bits • Hk+1(uk+1) becomes a random variable • Flow Entropy Equation: • Covert Information Channels • In data networks, information is transmitted by the contents, lengths and timings of packets only • Covert information can be conveyed by the lengths [1] and the timings of packets [2] • Side information considered in packet lengths only • Stochastic characterization of flow with maximum entropy • Notation • rk = token increment for the kth slot • Bk = bucket depth for the (k+1)th slot • lk = length of packet transmitted in kth slot • uk = residual tokens at start of the kth slot References [1] R.G. Gallager, “Basic Limits on Protocol Information in Data Communication Networks”, IEEE Transactions On Information Theory, vol. 22, pp. 385-398, July 1976. [2] V. Anantharam and S. Verdu, “Bits Through Queues”, IEEE Transactions on Information Theory, vol. 42, pp. 4-18, Jan. 1996. [3] S. Giordano and J.-Y.L. Boudec, “On a Class of Time Varying Shapers with Application to the Renegotiable Variable Bit Rate Service”, Journal on High Speed Networks, vol. 9, pp. 101-138, June 2000. [4] P. Shah and A. Karandikar, “Optimal Packet Length Scheduling for Regulated Media Streaming”, IEEE Communications Letters, vol. 7, pp. 409-411, August 2003. [5] P. Shah and A. Karandikar, “Information Utility of Token Bucket Regulator”, Electronics Letters, vol. 39, pp. 581-582, March 2003. • Optimal Flow Entropy Equation • Boundary condition: HN (uN ) = 0 • Probability constraint: • For a given GTBR, optimal probability sequences can be determined stage-by-stage backward recursively • Optimal Flow Entropy Equation: • Information utility of GTBR = H0*(0) • For more information • Email: adgore@ee.iitb.ac.in • URL: http://www.ee.iitb.ac.in/uma/~adgore