Opportunistic Scheduling with Limited Channel State Information: A Rate Distortion Approach Matthew Johnston, Eytan Modiano, Yury Polyanskiy Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA Email: fmrj, modiano, [email protected] Abstract—We consider an opportunistic communication The above minimization can be formulated as a rate system in which a transmitter selects one of multiple channels distortion optimization with an appropriately designed over which to schedule a transmission, based on partial distortion metric. The opportunistic communication frame- knowledge of the network state. We characterize a fun- damental limit on the rate that channel state information work, in contrast to traditional rate distortion, requires must be conveyed to the transmitter in order to meet a that the channel state information sequence be causally constraint on expected throughput. This problem is modeled encoded, as the receiver causally observes the channel as a causal rate distortion optimization of a Markov source. states. Consequently, restricting the rate distortion problem We introduce a novel distortion metric capturing the impact to causal encodings provides a tighter lower bound on the of imperfect channel state information on throughput. We compute a closed-form expression for the causal information required CSI that must be provided to the transmitter. rate distortion function for the case of two channels, as well Opportunistic scheduling is one of many network con- as an algorithmic upper bound on the causal rate distortion trol schemes that requires network state information (NSI) function. Finally, we characterize the gap between the causal in order to make control decisions. The performance of information rate distortion and the causal entropic rate- these schemes is directly affected by the availability and distortion functions. accuracy of this information. If the network state changes I. INTRODUCTION rapidly, there are more possibilities to take advantage of Consider a transmitter and a receiver connected by two an opportunistic performance gain, albeit at the cost of independent channels. The state of each channel is either additional overhead. For large networks, this overhead can ON or OFF, where transmissions over an ON channel become prohibitive. result in a unit throughput, and transmissions over an OFF This paper presents a novel rate distortion formulation channel fail. Channels evolve over time according to a to quantify the fundamental limit on the rate of overhead Markov process. At the beginning of each time slot, the required for opportunistic scheduling. We design a new receiver measures the channel states in the current slot, distortion metric for this setting that captures the impact and transmits (some) channel state information (CSI) to on network performance, and incorporate a causality con- the transmitter. Based on the CSI sent by the receiver, the straint to the rate distortion formulation to reflect practical transmitter chooses over which of the channels to transmit. constraints of a real-time communication system. We ana- In a system in which an ON channel and OFF channel lytically compute a closed-form expression for the causal are equally likely to occur, the transmitter can achieve an rate distortion lower bound for a two-channel system. 1 Additionally, we propose a practical encoding algorithm expected per-slot throughput of 2 without channel state in- 3 to achieve the required throughput with limited overhead. formation, and a per-slot throughput of 4 if the transmitter has full CSI before making scheduling decisions. How- Moreover, we show that for opportunistic scheduling, there ever, the transmitter does not need to maintain complete is a fundamental gap between the mutual information and knowledge of the channel state in order to achieve high entropy-rate-based rate distortion functions, and discuss throughput; it is sufficient to only maintain knowledge of scenarios under which this gap vanishes. Proofs have been which channel has the best state. Furthermore, the memory omitted for brevity. in the system can be used to further reduce the required II. PROBLEM FORMULATION CSI. We are interested in the minimum rate that CSI must be sent to the transmitter in order to guarantee a lower Consider a transmitter and a receiver, connected through bound on expected throughput. This quantity represents a M independent channels. Assume a time slotted system, fundamental limit on the overhead information required in where at time-slot t, each channel has a time-varying this setting. channel state Si(t) 2 fOFF; ONg, independent from all other channels. The notation Si(t) 2 f0; 1g is used This work was supported by NSF Grants CNS-0915988, CNS- interchangeably. 1217048, ARO MURI Grant W911NF-08-1-0238, ONR Grant N00014- 12-1-0064, and by the Center for Science of Information (CSoI), an NSF Let X(t) = Xt = fS1(t);S2(t);:::;SM (t)g represent Science and Technology Center, under grant agreement CCF-09-39370. the system state at time slot t. At each time slot, the p satisfying n n X X n n n n n 1 − p OFF ON 1 − q E[d(x1 ; z1 )] = p(x1 )q(z1 jx1 )d(x1 ; z1 ) ≤ D: n n x1 z1 q (3) where p(xn) is the PDF of the source, and the causality Fig. 1: Markov chain describing the channel state evolution of 1 each independent channel. constraint: i n i n n n i i q(z1jx1 ) = q(z1jy1 ) 8x1 ; y1 s.t. x1 = y1; (4) transmitter chooses a channel over which to transmit, with the goal of opportunistically transmitting over an ON Mathematically, the minimum rate that CSI must be trans- channel. Channel states evolve over time according to a mitted is given by Markov process described by the chain in Figure 1, with NG 1 n 1 1 Rc (D) = lim inf H(Z1 ) (5) n!1 transition probabilities p and q satisfying p ≤ 2 and q ≤ 2 , q2Qc(D) n corresponding to channels with “positive memory.” where 1 H(Zn) is the entropy rate of the encoded se- The transmitter does not observe the state of the system. n 1 quence in bits. Equation (5) is the causal rate distortion Instead, the receiver causally encodes the sequence of function, as defined by Neuhoff and Gilbert [1], and channel states Xn into the sequence Zn and sends the 1 1 is denoted using the superscript NG. This quantity is encoded sequence to the transmitter, where Xn is used to 1 an entropy rate distortion function, in contrast to the denote the vector of random variables [X(1);:::;X(n)]. information rate distortion function [2], [3], [4], which will The encoding Z(t) = Z 2 f1;:::;Mg represents the t be discussed in Section III. The decision to formulate this index of the channel over which to transmit. Since the problem as a minimization of entropy rate is based on the throughput-optimal transmission decision is to transmit intuition that the entropy rate should capture the average over the channel with the best state, it is sufficient for number of bits per channel use required to convey channel the transmitter to restrict its knowledge to the index of the state information. channel with the best state at each time. The expected throughput earned in slot t is E[thpt(t)] = B. Previous Work S (t) i = Z(t) Z(t) , since the transmitter uses channel , Among the earliest theoretical works to study commu- 1 and receives a throughput of if that channel is ON, nication overhead in networks is Gallager’s seminal paper and 0 otherwise. Clearly, a higher throughput is attainable [5], where fundamental lower bounds on the amount of with more accurate CSI, determined by the quality of overhead needed to keep track of source and destination Zn the encoding 1 . The average distortion between the addresses and message starting and stopping times are xn zn sequences 1 and 1 is defined in terms of the per-letter derived using rate-distortion theory. A discrete-time analog distortion, of Gallager’s model is considered in [6]. A similar frame- n 1 X work was considered in [7] and [8] for different forms of d(xn; zn) = d(x ; z ); (1) 1 1 n i i network state information. i=1 The traditional rate-distortion problem [9] has been where d(xi; zi) is the per-letter distortion between the ith extended to bounds for Markov Sources in [10], [11], source symbol and the ith encoded symbol at the transmit- [12]. Additionally, researchers have considered the causal ter. For the opportunistic communication framework, the source coding problem due to its application to real-time per-letter distortion is defined as processing. One of the first works in this field was [1], in which Neuhoff and Gilbert show that the best causal d(x ; z ) 1 − [thpt(t)] = 1 − S (t); (2) i i , E Z(t) encoding of a memoryless source is a memoryless coding, where SZ(t) is the state of the channel indexed by Z(t). or a time sharing between two memoryless codes. Neuhoff Thus, an upper bound on expected distortion translates and Gilbert focus on the minimization of entropy rate, as to a lower bound on expected throughput. Note that the in (5). The work in [13] studied the optimal finite-horizon traditional Hamming distortion metric is inappropriate in sequential quantization problem, and showed that the this setting, since the transmitter does not need to know optimal encoder for a kth-order Markov source depends the channel states of channels it will not transmit over. on the last k source symbols and the present state of the decoder’s memory (i.e. the history of decoded symbols). A. Problem Statement A causal (sequential) rate distortion theory was intro- duced in [3] and [14] for stationary sources.