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] Notation r k = token increment for the k th slot B k = bucket depth for the (k+1) th slot l k = length of packet transmitted in k th slot u k = residual tokens at start of the k th slot 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 = (r 0, r 1, …, r N-1 ) = token increment sequence B = (B 0, B 1, …, B N-2 ) = bucket depth sequence 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. Entropy-Optimal Generalized Token Bucket Regulator Ashutosh Deepak Gore and Abhay Karandikar Department of Electrical Engineering, Indian Institute of Technology - Bombay References [1] R.G. Gallager, “Basic Limits on Protocol Information in Data Communication Networks”, IEEE Transactions On Information Theory, vol. 22, pp , July [2] V. Anantharam and S. Verdu, “Bits Through Queues”, IEEE Transactions on Information Theory, vol. 42, pp. 4-18, Jan [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 , June [4] P. Shah and A. Karandikar, “Optimal Packet Length Scheduling for Regulated Media Streaming”, IEEE Communications Letters, vol. 7, pp , August [5] P. Shah and A. Karandikar, “Information Utility of Token Bucket Regulator”, Electronics Letters, vol. 39, pp , March For more information URL: 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 Flow Entropy Equation Information utility is the maximum entropy achievable by any flow which is constrained by the GTBR R(N, r, B) Entropy H k is a function of system state u k If a packet of length l k bits is transmitted with probability  overt information transmitted = l k bits  covert information = bits  H k+1 (u k+1 ) becomes a random variable Flow Entropy Equation: Optimal Flow Entropy Equation Boundary condition: H N (u N ) = 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 = H 0 * (0) Problem Statement Can a generalized token bucket regulator achieve higher information utility than a standard IETF token bucket regulator? Given STBR R s (N, r, B), determine r and B of GTBR R g (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