On the queueing delay of a multicast erasure channel Brooke Shrader and Anthony Ephremides Dept. of Electrical and Computer Engineering (ECE) and Institute for Systems Research (ISR) University of Maryland College Park, Maryland, U.S.A. {bshrader, etony}@umd.edu

Abstract — In this work we address the stability and delay performance of a multicast erasure channel with random arrivals at the source node. We consider both a standard retransmission (ARQ) scheme as well as random linear coding. Our results show that while random linear coding may outperform retransmissions for heavy traffic, the delay incurred by the use of random linear codes is significantly higher when the source is lightly loaded. I. Introduction Random linear network coding [1, 2] has recently emerged as a technique which improves upon simple forwarding strategies by increasing the amount of data delivered per time unit for multicast transmission. The analysis that leads to this result typically assumes that at any given time, there are multiple data packets awaiting transmission at a source and that random linear coding is performed over these packets. However, in a realistic network setting, the buffer of packets awaiting transmission at a source may sometimes empty. Similarly, the buffer may have only one packet awaiting transmission, and the source may choose to wait for additional packets to arrive in order to perform random linear coding. In order to understand such scenarios, in this work we analyze random linear coding when performed over packets which randomly arrive at the source node. We consider a single source node which multicasts to M destination nodes. The transmission can be carried out by one of two schemes. The first is a random linear coding scheme, in which random linear combinations of packets awaiting transmission at the source are sent continually until all destinations have received enough linear combinations to perform decoding. Alternatively, the source can continually retransmit uncoded packets until all destinations have received. We present a queueing model for these two schemes and compare their performance in terms of stability, or the number of packets per 1 This work is supported in part by the Office of Naval Research through grant N000140610065 and by the Department of Defense under MURI grant S0176941. Prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under The Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U. S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon.

λ

K ...

21 ...

Encoder

q1 q2 qM

Decoder 1 Decoder 2 . . . Decoder M

Figure 1: Packets arrive randomly to a single source node, are stored in an infinite-capacity buffer, and random linear coding is performed before multicast transmission to M destinations. slot that the source can process while maintaining finite delay, and in terms of the average delay. Our work can also be viewed as an analysis of random linear Fountain codes [3], and our model can be applied to analyze other forms of Fountain codes, such as LT codes [4]. The problem we consider is most closely related to the recent works on random linear coding at a source with a finite buffer for storing incoming packets [5] and on the delay performance for a source which always has packets awaiting transmission [6]. Our work differs from those and other works in a number of ways. First, we consider random arrivals to an infinite-capacity buffer at the source. Additionally, our model accounts for the situation in which a source node must wait for additional data packets to arrive before beginning the encoding process. Alternative approaches, such as beginning encoding and transmission as soon as a single packet has arrived (as is done in [5]), may reduce the waiting time for arriving packets, but a cost in terms of overhead may be incurred due to the need for additional feedback to the source for acknowledging successful transmissions. In our work we treat the number of data packets used in encoding, K, as a parameter. Finally, a number of previous works, including [5, 6], consider random linear coding over a finite field and perform analysis by letting the field size approach infinity. In contrast, our work deals with random linear coding over binary data and studies the delay obtained when transmitting packets of finite length. II. System model The system we consider is shown in Figure 1. Time is slotted; a slot corresponds to the time needed for a packet

to be transmitted over the channel. Data packets of finite, fixed length arrive to a source node through a Bernoulli process with rate λ packets/slot. Thus λ represents the probability that a data packet arrives in a slot. Upon arrival, the data packet is placed in a buffer of infinite capacity in order to await encoding and transmission. A newly arriving packet is placed at the end of the buffer behind other packets that arrived earlier. Before transmission, the data packets must undergo encoding, and the parameter K indicates the number of data packets utilized in the encoding process. If there are fewer than K data packets in the buffer, then those packets must await the arrival of additional packets before encoding and transmission can begin. This assumption will lead to a delay penalty for lightly loaded systems. Once there are K data packets in the buffer, the first K packets are simultaneously passed to the encoder and removed from the buffer. The encoder then forms random linear combinations of the data packets. We refer to the random linear combinations as coded packets and assume that given the set of K data packets, a coded packet can be generated instantaneously. Each of the K data packets is included in the random linear combination with probability 1/2. We assume that information on which of the K data packets is included in each coded packet is available to the decoder at a destination node. This can be accomplished either through the use of a packet header or through the use of random number generators with identical seeds at the source and destinations. Coded packets are transmitted from the source to a set of M destination nodes. The channel of each sourcedestination pair is an independent erasure channel with reception probability in each slot given by qm = Pr{coded packet received at destination m} (1) for m = 1, 2, . . . , M . The source node will transmit coded packets over the channel until enough coded packets are received for the destination to perform decoding. Let the e denote the number of coded packets random variable N needed at a destination in order to decode. As each dese tination may receive a different set of coded packets, N may take a different value for each destination in any ree will depend on the way alization. The distribution of N in which coded packets are formed. In the following sece and present tions, we will analyze the distribution of N results on random linear codes. Our methodology can also be applied to Fountain codes, such as LT codes [4], e. through varying the distribution of N e coded packWhen the destination node has received N ets, decoding is performed to uncover the original K data packets. We assume that decoding occurs instantaneously and simultaneously on all K data packets. Subsequently, an error-free acknowledgement message is instantaneously sent from the destination to the source. Once the source has received acknowledgements from all M destinations, it proceeds with encoding and transmission of the next K data packets in the front of its buffer.

In the following section we model the system shown in Figure 1 as a discrete-time queue and characterize the arrival and service processes. Much of the analysis is carried out through use of the probability generating function (pgf). For a non-negative discrete random variable with probability mass function (pmf) f (n), the pgf, denoted byPF (z), is the z-transform of f (n) and is given by F (z) = n≥0 f (n)z n . III. Queueing model for random linear coding

A

Arrival process

As stated previously, data packets arrive to the source according to a Bernoulli process with rate λ packets/slot. Accordingly, the time that elapses between arrivals of individual data packets is geometrically distributed. Let Tei denote the time in slots between the arrival of the i − 1st and the ith packet in the buffer. The pmf of Tei is given by ai (t) = Pr{Tei = t} = λ(1 − λ)t−1 , t ≥ 1 (2) and the pgf is given by Ai (z) =

λz . 1 − z(1 − λ)

(3)

Our arrival process must also account for the fact that newly arriving data packets must wait until the buffer contains K packets before encoding and transmission can begin. This scenario is referred to as bulk service and leads to the Erlangian interarrival time distribution [7] for continuous-time queues. The approach for our discretetime system is similar. We let Te denote the effective interarrival time, which is essentially the time in slots between the arrival of every K th packet. Thus Te is given by the sum of K independent random variables Tei . The pgf of Te, which we denote A(z), can be expressed as µ ¶K λz K A(z) = (Ai (z)) = . (4) 1 − z(1 − λ) By taking derivatives of A(z) with respect to z and evaluating at z = 1, we can find the mean and variance of the interarrival time as given below. E[Te] =

B

K , λ

K(1 − λ) σa2 = V ar[Te] = λ2

(5)

Service process

The service process characterizes the time that elapses between the K th data packet entering the encoder and receipt at the source of acknowledgement messages from e denote the service time, or the all M destinations. Let X e number of slots needed for all M destinations to receive N e coded packets. Furthermore, let Xm denote the number of slots needed for destination m, m = 1, 2, . . . , M , to e coded packets. Then we have receive N e = max X em . X m

(6)

Equation (6) accounts for the multicast nature of the transmission from the source. em through We proceed by deriving the distribution of X use of its pgf Bm (z). First, we consider the time that elapses between two consecutive successful receptions of coded packets. According to our erasure channel model, this time will be geometrically distributed with success probability qm in each slot. The corresponding pgf, which we denote Γm (z), is given by Γm (z) =

qm z . 1 − z(1 − qm )

(7)

The time periods that elapse between reception of consecutive coded packets will be independent, identically distributed according to Γm (z). Since the destination must e such coded packets, the service time to destinareceive N e independent random tion m will be given by the sum of N variables distributed as Γm (z). Letting FK (z) denote the e when K data packets are used in encoding, the pgf of N em can be shown (see, e.g., [7]) to be given by pgf of X Bm (z) = FK (Γm (z)).

(8)

e to obtain the pgf Next we find the distribution of N e FK (z). The distribution of N reflects the random linear coding, in which a coded packet is formed by a modulo-2 sum of a subset of the K data packets, where each data packet is included in the sum with probability 1/2. Let FK (n) denote the cumulative distribution function (cdf) e , or the probability that the number of coded packof N ets needed for decoding is less than or equal to n. With each coded packet it receives, the destination collects a binary column of length K in a matrix, where the column indicates which of the K data packets is included in the random linear combination for that particular coded packet. Then FK (n) is the probability that decoding can be performed (e.g., by Gaussian elimination) if the matrix has at most n columns. Thus

e as fK (n) = Pr{N e = n}, We now define the pmf of N the probability that decoding can be performed when the matrix has exactly n columns and no fewer. Thus fK (n) = FK (n) − FK (n − 1).

(12)

We have numerically computed fK (n) for various values of K and observed its behavior. The distribution fK (n) is nearly identical for all values of K > 10, which was also observed in [3]. Additionally, although fK (n) has infinite support, we observed that the first few terms contain nearly all of the probability mass. This characteristic will be useful in our numerical computations, in which we truncate the pmf fK (n). Before moving on, we make one comment regarding e . In deriving (11) we assumed that the distribution of N nK there are 2 possible realizations of a random binary K × n matrix. As such we allow the possibility that a column of the matrix is all-zero, or equivalently, that a coded packet is formed as the modulo-2 sum of 0 data packets. Clearly, transmitting the sum of 0 data packets would be wasteful and avoided in a practical system. We allow all-zero columns in our model due to the resulting ease in computation. We note that for large values of K the probability of generating an all-zero column is very small. We observed that for K ≥ 10 the effect of all-zero columns on fK (n) is negligible. Using the expression in (8), we can write the pgf of em as follows. X µ ¶n ∞ X qm z Bm (z) = fK (n) (13) 1 − z(1 − qm ) n=K

em , which we denote bm (xm ), is found by The pmf of X taking an inverse z-transform and is given by (P ∞ n xm −n , xm ≥ K n=K fK (n)βmn qm (1 − qm ) bm (xm ) = 0, xm < K (14) where

(xm −1)(xm −2) . . . (xm −(n−1)) FK (n) = Pr{a random K×n binary matrix has rank K}. βmn = . (15) (n − 1)! (9) Note that for n < K the matrix cannot possibly have We will make use of bm (xm ) as shown above in our numerrank K and FK (n) = 0. For n ≥ K, we can write an ical computations. Finally, we can find the distribution expression for FK (n) by first counting up the number of e = maxm X em using the distribution of X em . Letting of X K × n non-singular matrices, which is given by e we have B(x) denote the cdf of X, (2n − 1)(2n − 2)(2n − 22 ) . . . (2n − 2K−1 ).

(10)

In the product above, each term accounts for a row of the K × n binary matrix and reflects that the row is neither zero, nor equal to any of the previous rows, nor equal to any linear combination of previous rows. Since the total number of K × n binary matrices is 2nK , we obtain FK (n) =

K−1 Y i=0

(1 − 2−n+i ),

n ≥ K.

(11)

e1 ≤ x} × Pr{X e2 ≤ x} × . . . × Pr{X eM ≤ x}. B(x) = Pr{X (16) e which we denote E[X] e and The mean and variance of X, σb2 respectively, can be found from B(x) and are used in analyzing the stability and delay.

C

Stability and delay

We next consider the conditions for stability of the queue, or more formally, the conditions for which the Markov

chain representing the waiting time in the queue is ergodic. We define the traffic intensity ρRLC as e λE[X] . K

(17)

For a standard queue, the queue is stable if and only if ρRLC < 1 [7]. If this condition is not satisfied, then the delay will grow without bound. In examining the delay, we note that the interarrival and service time distributions in the queueing model developed above do not allow for the application of standard delay results, such as the delay for the M/G/1 queue [7]. Instead, we will make an analytical approximation to the delay based on the mean and variances of the interarrival and service processes. In doing so, we note that the delay will consist of two terms: the service time and the waiting e and is time. The expected service time is given by E[X] constant over all arrival rates λ. The waiting time can be attributed to two phenomena: the time spent waiting to reach the front of the queue while other groups of K packets are being served, and the time spent waiting for additional packets to arrive to form a group of K packets for encoding and transmission. For the time spent waiting to reach the front of the queue, we use the heavy traffic approximation provided for continuous-time systems, which is given in [8] by σa2 + σb2 . 2E[Te](1 − ρRLC )

(18)

This quantity will be relatively small for small values of λ but will be the dominant term as λ increases, approaching infinity as ρRLC → 1. For the time spent waiting for additional packets to arrive, we note that on average, a given packet will need to wait for (K − 1)/2 additional packets to arrive, which will require a waiting time of K−1 2λ . This quantity will approach infinity as λ → 0 but will diminish as λ increases. Our delay approximation assumes that the waiting time will be given by the maximum of the two quantities described above. Thus, our approximation for the average delay is given by à ! 2 2 σ + σ K − 1 a b e + max DRLC ≈ E[X] , . 2E[Te](1 − ρRLC ) 2λ (19) IV. Queueing model for retransmissions The retransmissions scheme behaves similarly to the random linear coding scheme for K = 1. Arrivals form a Bernoulli process with expected interarrival time 1/λ, which corresponds to E[Te] in the random linear coding model with K=1. The service time, which we deeRE , is the maximum of M geometrically disnote X tributed random variables with success probabilities qm , eRE m = 1, 2, . . . , M in each slot. We compute the cdf of X as eRE ≤ x} = Pr{X f0 1 ≤ x}×. . .×Pr{X f0 M ≤ x} (20) Pr{X

q =0.9, q =0.9, q =0.9

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Stable arrival rate λ (packets/slot)

ρRLC =

1

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Figure 2: The maximum stable arrival rates for random linear coding over K packets (dashed line) and retransmissions (solid line) for four different channel models and transmission to M =3 destination nodes. f0 m is geometrically distributed with success probwhere X ability qm in each slot. From this distribution, we can find eRE , which we denote the first and second moments of X 2 e e E[XRE ] and E[XRE ], respectively. The traffic intensity eRE ]. The necessary and sufρRE is given by ρRE = λE[X ficient condition for stability is ρRE < 1. For the retransmission scheme, the queueing model fits the description of a Geo/G/1 queue, for which exact delay results are available in [9]. The average delay is given by e2 eRE ] + λE[XRE ] − ρRE . DRE = E[X 2(1 − ρRE )

(21)

V. Numerical results For our numerical results we have computed probability distributions, including fK (n) and bm (xm ), to within 10−5 of the total probability mass. The maximum stable arrival rates for random linear coding and retransmissions are compared in Figure 2 for four different sets of reception probabilities (q1 , q2 , q3 ). These results indicate that when one of the destinations has a poor channel (q3 =0.3) there is little or no benefit of random linear coding over retransmissions in terms of the stable arrival rate. Also, even for relatively good channels, the random linear coding scheme does not unconditionally outperform the retransmission scheme: if the value of K is not sufficiently large, then the retransmission scheme can support higher average arrival rates. The effect of the number of destination nodes on the stable arrival rate is shown in Figure 3. These results are computed for relatively good channels in which qm = 0.8, m = 1, 2, . . . , M , and random linear coding nearly always outperforms retransmissions. The arrival rate that the source can tolerate while maintaining stability is higher when the source multicasts to fewer destinations. Additionally, we observe that random linear coding provides larger gains over retransmissions when the source multicasts to more destination nodes.

0.8

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Retransmissions K=10,Approx K=10,Sim K=40,Approx K=40,Sim

250 0.7

M=2,RLC M=2,Retrans. M=5,RLC M=5,Retrans. M=10,RLC M=10,Retrans.

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Figure 3: The maximum stable arrival rates for random linear coding over K packets (dashed line) and retransmissions (solid line) for channels with qm = 0.8, m = 1, 2, . . . , M and different numbers of destinations. The delay performance of random linear coding is compared with retransmissions in Figure 4 for a channel with (q1 , q2 , q3 ) = (0.9, 0.9, 0.9). In addition to computing DRLC and DRE , we have performed and plotted the results of a Monte Carlo simulation of the random linear coding scheme over 20,000 slots in order to compare the outcome with our approximation. Our approximation closely matches the simulation results for small values of λ, however, the two outcomes differ when the queues saturate. We note that, in contrast to typical queueing delay results, the delay for random linear coding is not a monotonic increasing function. The behavior of the random linear coding delay for λ → 0 is due to the time spent waiting for additional packets to arrive in order to form a group of K packets for encoding and transmission. For random linear coding over K=10 packets, the delay is higher than the retransmission delay for all values of λ. The fact that the random linear coding scheme with K=10 saturates sooner than the retransmission scheme reflects the result shown in Figure 2. Thus, for M =3, K=10, and this particular channel model, random linear coding offers no benefit over retransmissions in terms of neither stability nor delay. The random linear coding scheme over K=40 packets can tolerate higher values of λ than the retransmissions scheme, however, for small values of λ, the random linear coding scheme with K=40 provides significantly higher delay than retransmissions. This result points to a tradeoff between the stable throughput and the delay. VI. Conclusion We have compared the use of random linear coding to retransmissions for multicast transmission from a source at which packets arrive randomly. Our results indicate that random linear coding does not outperform retransmissions in terms of stable arrival rate when one of the destinations has a very poor channel. However, we also observed that when the channel is relatively good, random linear coding provides benefits which grow with the

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Figure 4: Delay versus arrival rate λ for retransmissions (solid line), the random linear coding approximation (dotted line), and the random linear coding simulation (o,+). Random linear coding is performed over K = 10 (o) and K = 40 (+) packets. The results are for transmission to M =3 destinations with (q1 , q2 , q3 ) = (0.9, 0.9, 0.9). number of multicast destinations. In terms of delay, although the retransmission delay may saturate sooner than random linear coding, the coding scheme incurs significantly higher delay for small arrival rates. Our results document a tradeoff between stable throughput and delay when comparing random linear coding to retransmissions. Acknowledgments The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U. S. Government. References [1] S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. Inform. Theory, vol. 49, pp. 371-381, February 2003. [2] T. Ho, R. Koetter, M. Medard, D. R. Karger, and M. Effros, “The benefits of coding over routing in a randomized setting,” IEEE International Symposium on Information Theory, 2003. [3] D. J. C. MacKay, “Fountain codes,” Fourth IEE Workshop on Discrete Event Systems (WODES98), 1998. [4] M. Luby, “LT codes,” IEEE Symposium on the Foundations of Computer Science, pp. 271-280, 2002. [5] D. S. Lun, P. Pakzad, C. Fragouli, M. Medard, and R. Koetter, “An analysis of finite-memory random linear coding on packet streams,” Fourth International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt06), April 2006. [6] A. Eryilmaz, A. Ozdaglar, and M. Medard, “On delay performance gains from network coding,” Conference on Information Sciences and Systems (CISS), 2006. [7] L. Kleinrock, Queueing Systems, Volume I: Theory, New York: Wiley, 1975. [8] L. Kleinrock, Queueing Systems, Volume II: Computer Applications, New York: Wiley, 1976. [9] H. Takagi, Queueing Analysis, Volume 3: Discrete-time Systems, Amsterdam: North-Holland, 1993.