Capacity Region of the Permutation Channel John MacLaren Walsh and Steven Weber Abstract— We discuss the capacity region of a de- graded broadcast channel (DBC) formed from a chan- nel that randomly permutes input packets by selecting a permutation according to a probability distribution. Starting from the known capacity region expression for the DBC, we give an explicit outer and inner bound to the capacity region which are shown to be equal for the cases of 2 and 3 packets. The work extends previous results which considered the case where the permutation was selected uniformly from the set of all permutations. The results are useful in determining fundamental rate delay tradeoffs when transmitting temporally ordered content over multipath routed networks. I. I NTRODUCTION Consider a point to point channel that randomly permutes a set of packets applied at the input, which we call a permutation channel. In particular, the in- put is an ordered set of M different K-bit packets, (x 1 ,...,x M ), and the output is the packets in permuted order, (x π(1) ,...,x π(M) ) (the bits in the packets are unaffected by the channel). The likelihood of each possible permutation is specified by a probability distri- bution on all possible M ! permutations. The M packet arrival instants at the destination correspond to M receivers in a degraded broadcast channel (DBC). In particular, we consider the channel to have M outputs, with the mth ouput consisting of the all of the packets that have arrived until just after the arrival of the mth received packet at the sink. Although not necessary, it is convenient, and in our opinion natural, to assume the packets are labeled from 1 to M . Note that the total overhead of these labels, ⌈log 2 M ⌉ bits per packet, is amortized through the use of packets of long length, K. The channel is clearly degraded as the information available after m packet arrivals is a strict subset of the information available after m +1 arrivals. In this paper, we seek a closed form description for the capacity region of this DBC. Both authors are with the Department of Electrical and Computer Engineering of Drexel University. The contact author is J. Walsh (jwalsh@ece.drexel.edu). Manuscript date: October 17, 2008. A. Motivation: Delay Mitigating Codes for Multipath Routed Networks A major motivation for studying the permutation channel is the problem of using coding to mitigate delay in multipath routed networks. In order to make the correspondence between a simple model for this problem and the permutation channel, consider a net- work in which a source wishes to communicate with a sink, and has a diversity of M independent network paths through which it can communicate with the sink. Because there are other packets in the queues along these paths from other independent flows, M packets transmitted at the same time from the source along the M independent paths will arrive at the destination in a random order at different times. Suppose also that the information that the source wishes to transmit to the sink is temporally ordered. This could occur, for instance, if the source is streaming a multimedia clip to the sink composed of frames of multimedia data corresponding to different evenly spaced time instants. Alternatively, the source could be sending a sequence of control instructions to the sink through the network which must be executed in order, and within a short amount of time after transmission (e.g., to ensure stability of the associated control loop). In both instances, the unknown order of arrival of the M packets introduces ambiguity as to when data will be available for use at the sink. The purpose of delay mitigating codes is to encode the contents of the packets sent along the M paths in such a manner as to allow for successive decoding of information at the receiver at the schedule dictated by the content (i.e., playing out the multimedia or control frames when they need to be played). Such delay mitigating codes must exhibit a rate-delay tradeoff. A low delay guarantee requires extra coding redundancy, which in turn limits the number of distinct source data bits, which lowers the channel rate. Our initial work [1], [2], [3] has characterized this rate delay tradeoff when the permutation distribution is uniform. In this instance, it is possible to build codes that achieve points on the boundary of the capacity Forty-Sixth Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 23-26, 2008 WeD4.1 978-1-4244-2926-4/08/$25.00 ©2008 IEEE 646