Random Linear Network Coding for Time-Division Duplexing: Queueing Analysis

We study the performance of random linear network coding for time division duplexing channels with Poisson arrivals. We model the system as a bulk-service queue with variable bulk size. A full characterization for random linear network coding is provided for time division duplexing channels [1] by means of the moment generating function. We present numerical results for the mean number of packets in the queue and consider the effect of the range of allowable bulk sizes. We show that there exists an optimal choice of this range that minimizes the mean number of data packets in the queue.

💡 Research Summary

This paper investigates the performance of random linear network coding (RLNC) over time‑division duplex (TDD) channels when data packets arrive at a source according to a Poisson process. Because a TDD link can either transmit or receive at any instant, the transmitter must send a batch of coded packets back‑to‑back, then wait for an acknowledgment (ACK) before proceeding. The number of coded packets sent in each round (denoted N_i) depends on how many degrees of freedom (dof) the receiver still needs, which leads to a Markov‑chain description of the transmission process.

The authors model the system as a bulk‑service queue M/G(m,K)/1, where m and K are the minimum and maximum allowable batch sizes. When the buffer contains fewer than m packets the server waits; when it contains more than K packets it serves exactly K; otherwise it serves the exact number present. The service time is not constant but depends on the batch size, a feature captured by the moment‑generating function (MGF) of the completion time. Using the Markov‑chain structure, they derive a recursive expression for the MGF M_{T,n}(s) (Eq. 3) and show that the probability a^{(j)}k of observing k arrivals during a service of type j can be obtained from the MGF via A^{(j)}(z)=M{T,j}(λ(z−1)).

Stability requires λ < K·μ_K, where μ_K is the mean service rate for the largest batch. In the stable regime the stationary distribution π_i of the number of packets in the buffer can be expressed through the generating function Π(z). Π(z) involves the roots of A^{(K)}(z)−z^K, leading to a system of K linear equations plus a normalisation condition. Because solving for the roots can be numerically delicate, the authors also present a finite‑capacity version (capacity B) where the transition matrix is (B+1)×(B+1) and π can be obtained by solving a simple linear system.

Numerical experiments use realistic parameters: packet erasure probability 0.2, propagation delay 12.5 ms, 10 kbit data packets, header 80 bits, coding coefficient size 20 bits, link rate 1.5 Mbps, ACK size 100 bits, and N_i chosen to minimise mean transmission time as in the original RLNC‑TDD work. For λ=30 packets/s, K=5 and B=30, the stationary distribution is plotted for m=1,3,5, showing that smaller m concentrates probability mass on low queue lengths.

Tables I and II report the mean queue size E