Iterative Collision Resolution for Slotted ALOHA: An Optimal Uncoordinated Transmission Policy Krishna R. Narayanan and Henry D. Pfister Department of Electrical and Computer Engineering, Texas A&M University Email: {krn,hpfister}@tamu.edu Abstract—We consider a multi-user wireless network in which each user has one packet of information to transmit to a central receiver. We study an uncoordinated paradigm where the users send their packet a random number of times according to a prob- ability distribution. Instead of discarding the collided packets, the receiver performs iterative collision resolution. Recently, a few studies have shown that the iterative collision resolution process can be viewed as message-passing decoding on an appropriately defined Tanner graph. Using this equivalence, they used standard techniques to numerically optimize the probability distribution and demonstrated substantial throughput improvement over slotted ALOHA. In this paper, we show that the well-known soliton distribution is an optimal probability distribution and that the resulting throughput efficiency can be arbitrarily close to 1. Index Terms—Multiple-access, Collision resolution, Rateless Codes, Iterative Decoding I. I NTRODUCTION A. Problem Statement We consider a multi-user system with K users where each user wishes to transmit one packet of information to a central receiver. The total time available for communication is split into M time slots and the duration of each time slot is assumed to equal the time required to transmit one packet. In the j th time slot, a subset of users transmit their packets. There is no coordination between the users and each user independently uses a policy that specifies whether or not they will transmit their packet in the j th time slot. We assume that the receiver knows the exact set of users who transmit during every time slot. There are many ways that this information can be shared with the receiver but this is not the focus of the paper and, hence, not discussed in detail. Specifically, we consider a paradigm which is similar to the one in [1], where the kth user generates a random variable D k ∈{1,...,M } according to a probability mass function f D , i.e., Pr(D k = i)= f D [i]. Then, the user chooses D k time slots uniformly at random without replacement from the set {1,...,M } and transmits during these slots. In any given time slot, if exactly one user transmits a packet, then this packet is assumed to be decoded correctly. This is reasonable if good channel codes are used separately by This material is based upon work supported, in part, by the National Science Foundation (NSF) under Grant No. 0830696. Any opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsor. each user. If more than one user transmits their packet in the same time slot, a collision results. In this scheme, the receiver subtracts all the previously decoded packets from the collided packets and if the receiver is able to subtract all but one of the users, then a single-user decoder can be used to recover the last user’s packet. Otherwise, the received packet is saved in a buffer for future processing and every time a new packet is decoded, it is subtracted from all the packets in the buffer. We will refer to this as iterative interference cancellation since this is similar to interference cancellation in multi-user detection. When the receiver has received all the K packets correctly, an acknowledgment is sent to terminate the transmission. Since a total of M time slots is required to successfully transmit all K packets of information, the throughput of the system is said to be η = K/M packets/slot. Clearly, an upper bound on the throughput is η =1 packet/slot. The main result in the paper is to derive the optimal probability mass function f D for which this upper bound is achievable in the limit of K →∞, even when there is no coordination between the transmitters. B. Background It is well known that a standard slotted ALOHA scheme achieves a throughput efficiency of 1/e ≈ 0.37 and, hence, the proposed scheme is substantially better than standard slotted ALOHA. Recently, there have been a number of papers that consider collision resolution through iterative interference cancellation and thereby provide improved performance over slotted ALOHA. In [2] and [3], the time asymmetry between transmissions is exploited to bootstrap the iterative interference cancellation process. In the problem we consider, there is no time asymmetry and, hence, their results do not apply directly. The work that is most closely related to this paper is the work of Liva in [1] where a similar scheme as proposed here was considered and the author showed that an η of 0.965 can be achieved based on numerically optimizing the distribution f D . In [4], the authors consider an extension of the work in [1], where the nodes encode their packet before transmission and, again, numerically optimize the distribution. However, the question of whether or not the upper bound on η ≤ 1 is achievable is not addressed. The main observation of this paper is that the optimal distribution for this problem is the dual of the well-known soliton distribution [5] and that one can get arbitrarily close to the upper bound η ≤ 1.