On the Delay and Throughput Gains of Coding in Unreliable Networks
A. Eryilmaz, A. Ozdaglar, M. M´edard, E. Ahmed Laboratory for Information and Decision Systems
Massachusetts Institute of Technology Cambridge, MA, 02139
Email:{eryilmaz, asuman, medard, ebad}@mit.edu
Abstract— In an unreliable packet network setting, we study the performance gains of optimal transmission strategies in the presence and absence of coding capability at the transmitter, where performance is measured in delay and throughput. Al- though our results apply to a large class of coding strategies including Maximum Distance Separable (MDS) and Digital Fountain codes, we use random network codes in our discussions because these codes have a greater applicability for complex network topologies. To that end, after introducing a key setting in which performance analysis and comparison can be carried out, we provide closed form as well as asymptotic expressions for the delay performance with and without network coding. We show that the network coding capability can lead to arbitrarily better delay performance as opposed to traditional strategies.
I. INTRODUCTION
There has been a growing interest in developing new transmission strategies for efficient use of scarce resources in wireless networks. This is mainly motivated by emerging bandwidth intensive applications such as downloading video or music files, which involves transmission of files to multiple (potentially heterogeneous) receivers. While the standard ap- proach to data transmission builds on the scheduling approach, where information is transmitted to one of multiple receivers as a function of their channel conditions, it has been recognized that broadcasting to multiple receivers using network coding may improve performance in such settings. A fundamental question is to understand and quantify the performance gains obtained from network coding in wireless networks.
There has been considerable effort in revealing various gains of network coding. For example, in recent works [3, 20], it has been shown that network coding provides significant buffer savings over traditional methods. Most of the existing research to date has focused on throughput gains obtained from network coding (c.f. [1, 12, 11, 13]). Although these throughput gains may appear to imply gains in delay through Little’s law, this is not clearly the case since coding is performed over large blocks and each packet in the block must await the completion of the whole block before it can be decoded. To capture these effects, one must study the system at the packet level instead of using the flow-level formulation of delay (see e.g. [24]). Despite considerable practical interest in the use of network coding in wireless communication systems, gains in delay performance resulting from network coding relative to traditional scheduling have not been analyzed or quantified.
In this paper, we develop a model to study delay perfor- mance of network coding and traditional scheduling strategies in unreliable networks. To that end, we consider a scenario where a sequence of incoming files to a transmitter are to be communicated over the time-varying wireless medium to a set of neighboring receivers. This model not only captures the cellular and satellite downlink communications, but also serves as a building block for the operation and analysis of multi-hop wireless networks, as will be discussed in this paper.
We assume that files are broadcast to the receivers in a rateless fashion, i.e., the subsequent transmissions do not start before the whole of the current file is received by all the interested receivers. Our goal is threefold. First, we identify the optimal strategies under two transmission modes, namely scheduling and network coding, and quantify and compare the delay performance. Second, we use this model to investigate the sensitivity of the delay gains of network coding to key system parameters such as the number of receivers in the system and the file size. Third, we show how these results can be extended to more general network settings.
Our model involves transmission of (multiple) files from a single transmitter to multiple receivers with varying channel conditions. The varying channel conditions are modeled as stochastic changes in ON/OFF state of the channel. We an- alyze the model both when Channel Side Information (CSI) about the state of receiver channels is available to the base station and when transmission must be carried without such information.
First part of our analysis focuses on the key scenario of transmitting a single flow, where a flow is a sequence of files generated according to a random process, and destined to the same set of receivers. We consider a dynamic traffic model, in which the files associated with the flow arrive according to a Poisson process. As a measure of performance, we first focus on the mean value of the completion time, which is defined as the time required to transmit all packets of the head-of-line file to all the receivers. For this metric we establish the following results: For the network coding mode, we show that the random linear coding introduced by Ho et al.[8] is optimal, in the sense of minimizing the mean completion time, both with and without CSI. This is interesting because it provides simple transmission strategy with no requirement of feedback, but still achieve optimal performance. For the scheduling mode, the presence of CSI affects the optimal strategy. While without
the CSI, the optimal scheduling policy is the Round Robin (RR), the characterization of the optimal policy in the presence of CSI necessitates a dynamic programming formulation, which we provide in the paper. Since the computation of the optimal policy using this formulation becomes intractable as the size of the problem increases, we also present an efficient heuristic policy which we use for numerical comparisons.
Our numerical analysis shows that network coding leads to a significant improvement in mean completion time with respect to scheduling both with and without CSI.
As a complementary measure of performance of the system, we consider the mean value of the waiting time of an incoming packet, which is defined as the average time between a typical file’s arrival and the completion of its service. It known from queueing literature that the mean waiting time is a function of the first and second moments of the completion time. For random linear coding, we provide closed-form expressions for the first and second moments of the completion time. However since these expressions are in terms of infinite-sums, they do not enable us to do sensitivity analysis with respect to system parameters. We therefore provide asymptotic approximations to the first and second moments which highlight the explicit dependence on key parameters. These asymptotic expressions for the moments of mean service time with network coding are new and should be of independent interest in the analysis of coded-networks.
For the RR scheduler, we present bounds on the first and second moments of the waiting time. These results allow us to study asymptotic gains of network coding compared to scheduling and establish a number of sensitivity results. In particular, our analysis shows the delay and throughput gains of network coding compared to scheduling as a function of file size and the number of receivers. Our analysis proves that in the dense network setting where the number of receivers is large, achievable throughput of network coding relative to scheduling scales linearly with the file size, while the mean waiting time of scheduling relative to network coding for the same load scales quadratically with the file size.
In the second part of our analysis, we focus on another canonical scenario where multiple streams are downloaded to a different set of receivers. For this scenario, we present the optimal transmission strategies under both scheduling and coding modes. We establish the following results: we show that a variant of the Longest Connected Queue (LCQ) policy introduced in [21] is the optimal network coding strategy;
we prove that coding across sessions (intersession network coding) is not favorable for our system; we characterize the optimal scheduling strategies both with and without CSI, and observe significant gains from network coding when CSI is not available. These findings are important in identifying the optimal methods to be employed when multiple flows are to be served by the transmitter.
Our paper differs from the existing work in this area by explicitly modeling delay performance in file downloads and allowing for transmission without CSI. Previous research has instead focused on either optimal scheduling with time-varying channel conditions (see [21, 22]), or on the capacity gains from network coding (see [15, 9, 10, 18, 14, 23]) under
various different scenarios. This work builds on the findings of [5], which provided the first quantification of delay gains of network coding by using mean service time as the performance metric. In a more recent independent work [6], Ghaderi et al. provided an asymptotic formulation of the mean delay gains by building on the work of Grabner et al. ([7]). In this work, we extend these results by considering dynamic arrivals and studying the mean waiting time to provide exact as well as asymptotic expressions for the gains of network coding and scheduling. In another independent work [16], Nguyen et al. study the network coding performance in a single-hop broadcast setting in the presence of acknowledgements from the receivers.
The first part of our work is most closely related to [19], where the authors study a multicast scenario with stochasti- cally arriving packets and provide a transform-based analysis of delay for arbitrary coding window sizes. This approach, while providing explicit characterizations of the distributions of the arrival and service processes, does not reveal the rela- tionship between the delay performance and the critical system parameters such as the number of users and the coding window size. Yet, characterization of such a relationship is important in understanding the impact of essential system parameters on performance, and hence in providing valuable insights for the design of efficient systems. In this paper, we exert considerable effort and utilize completely different machinery such as Mellin transforms to obtain an asymptotically accurate formulation of delay with respect to the number of users, channel statistics, and the coding window size. Also, different from [19], we study the scenario of multiple unicast sessions and discuss ways of extending the analysis to the multi-hop network setting.
The rest of the paper is organized as follows: In Section II, the system model is introduced along with the transmission modes of interest and our goals. In Section III, the key scenario of a single flow destined to all the receivers is analyzed in detail. Section IV considers the case of multiple unicast flows. A method to extend the results to more general network settings is suggested in Section V. Finally, we provide a summary and our concluding remarks in Section VI.
II. SYSTEM MODEL ANDGOALS
In this section, we describe the single-hop setting of one transmitter broadcasting to multiple receivers over indepen- dently time-varying channels. This setting not only models the characteristics of cellular or satellite systems, but also serves as the fundamental building block for more general networks.
The connection to general topologies will be made explicit in Section V.
a) Single-hop Setting: Consider a single transmitting node and a setN of receiving nodes that are connected to it over time-varying channels. We assume a time-slotted system to which all the nodes are assumed to be synchronized. The duration of each time slot is selected with respect to the coherence time of the associated system so that channels stay constant within each slot, and vary across time-slots.
A setF of flows generates a sequence of files to be multi- cast to a subset of the receivers. Specifically, Flowf ∈ F is
demanded by the set Nf ⊆ N of receivers1. Files associated with each flow arrive at (or are generated by) the transmitter according to a stochastic process. Each file associated with a flow f is composed of Kf packets, each of which is a vector of length m over a finite field Fd. We assume that the duration of a time-slot can accommodate a single packet. The files of each flow are accumulated in a separate queue2 to be transmitted in a First-In-First-Out (FIFO) manner. We assume that transmission of a file starts only after the transmission of the file prior to it in the queue is complete.
The channel between the transmitter and theith receiver is a randomly varying ON/OFF channel. We let Ci[t]∈ {0, 1}
denote the state of user i’s channel in slot t. We assume that Receiver-i successfully receives the packet transmitted at slot t if Ci[t] = 1, and it cannot receive anything if Ci[t] = 0. We will take each Ci[t] to be a Bernoulli random variable with mean pi that are independent across time and across receivers. The channels of different receivers can in general be asymmetric, i.e., pi may be different for different i∈ {1, · · · , N}. However, in parts of the subsequent analysis we will restrict our attention to the symmetric case ofpi= p for all i in order to have tractable formulations. The system model is depicted in Figure 1.
.. . C [t]N
C [t]1
C [t]2 ..
.
.. .
Trans- mitter HOL file of flow f (K packets)f
.. . .. .
Receiver 1
Receiver 2
Receiver N Pf,2
. . .
Pf,1 Pf,Kf
Flow f
Virtual queue for files of flow f
Fig. 1. System model
b) Availability of Channel Side Information (CSI): We distinguish between two cases regarding the availability of CSI at the transmitter. We say that CSI is available when the vector3 of channel states C[t] , (C1[t],· · · , CN[t]) is known by the transmitter at the beginning of slot t so that transmissions can be decided with the perfect knowledge of which receivers will get them. Such an assumption requires extra overhead for estimation and feedback operations, and may be impractical especially when the number of receivers is large or the channel variations are too fast to accommodate the feedback delay. We study this scenario under the assumptions of perfect and instantaneous feedback with negligible overhead as a limiting idealistic case. The outcome of this study will allow us to identify the strengths of weaknesses of different strategies even when CSI is available.
The more realistic scenario of NO-CSI refers to the case when no channel quality information is available to the trans- mitter at the outset of transmission. Thus, the decision as
1We will useF, N and Nf to denote the cardinalities of the sets F, N andNf, respectively.
2This queue need not be a physically separate buffer, but a virtual one where files of different flows are accounted for separately.
3We will consistently use boldface letters to denote vectors.
to what to transmit must be made blindly. In this extreme, we assume that feedback is very costly, and hence must be minimized. This is a reasonable assumption when the number of receivers is large. Thus, instead of intermediate scenarios such as ARQ-type schemes, we focus on the case where the receivers send acknowledgement (ACK) packets only when they receive the whole file. Thus, we assume a file-based ACK scheme, rather than the significantly more costly packet-based ACK scheme.
c) Transmission Strategies: The strategy employed by the transmitter to broadcast the head-of-the-line file to the receivers has a critical effect on the service time distribution of the file completion. We focus on two modes of transmission in this paper, namely scheduling and coding. Before we define these two modes, we introduce some notation. Since the files are transmitted in a FIFO order, we can focus on the head-of- line (HOL) file of flowf , which is composed of Kf packets.
Then, Packet-k of the HOL file of flow f is referred to as Pf,k, which is a vector of length m over a finite field Fd. Finally, let P[t] denote the packet chosen for transmission in slott.
Definition 1 (Scheduling). Schedulingrefers to the mode of transmission where in any given slot, the transmitter must pick a single packet from the HOL file to transmit. Specifically, we haveP[t]∈ {Pf,k|f ∈ F, k = 1, · · · , Kf} .
Definition 2 ((Network) Coding). (Network) Coding refers to the mode of transmission where in every slot, sayt,any linear combination of the packets belonging to the HOL file can be transmitted. Specifically, we have
P[t] =X
f∈F Kf
X
k=1
af,k[t]Pk,f,
whereaf,k[t]∈ Fdfor eachf ∈ Fandk∈ {1, · · · , Kf}.The transmitter chooses the coefficients{af,k[t]}for eacht.
d) Goals: For the above model, we are interested in
• identifying the optimal transmission strategies under scheduling and coding transmission modes, and under the assumption of CSI and NO-CSI, where the optimal strategy is the one which minimizes the mean service time;
• providing an analytical expression of the mean waiting time (including queueing delay and service time) for the incoming packets under the optimal transmission schemes;
• understanding the asymptotic effect of the number of users and the file sizes on the mean waiting time;
• providing methods for extending the single-hop setting to multiple-hop networks with general topologies.
We will address each of these goals in the subsequent analysis.
The rest of the paper is organized as follows. In Section III, we focus on the broadcast scenario where the incoming files are to be transmitted to all the receivers. After characterizing the optimal transmission strategies, we provide explicit as well as asymptotic expressions for their performance and show the significant gains of coding with respect to scheduling.
In Section IV, we consider the scenario of multiple unicast
flows, where each flow is destined to a separate receiver, and characterize the optimal policies in the two transmission modes. In Section V, we provide a method whereby the results and analysis of the single hop setting can be extended to multi- hop networks. Finally, we provide our concluding remarks in Section VI.
III. BROADCASTING ASINGLEFLOW
In this section, we focus on the key scenario of the transmit- ter broadcasting the incoming files of a single flow to all the receivers, i.e., we setF = 1, andNf =N in our model. This scenario allows us to isolate the delay analysis from issues of scheduling transmissions across flows, and allows for tractable analysis. Since there is only one flow in the system, we will drop the subscript f in our notation throughout this section, and denote Packet-k of the HOL file of the flow as Pk, and the size of the file as K.
We assume that files of the flow arrive according to a Poisson process4 of rate λ. The Poisson assumption allows us to view the whole system as an M/G/1 queue, where the service time distribution is a function of the transmission strategy being employed at the transmitter. Let Z(N, K) denote the time required to transmit all the packets of the HOL file to all the receivers under a given transmission strategy, and (N, K) parameters. We refer to Z(N, K) as the completion (or service) time of a file download. The mean waiting time W (λ, N, K) of an incoming file is given by the Pollaczek- Khinchin formula ([2]):
W (λ, N, K) = λE[Z(N, K)2]
2(1− λE[Z(N, K)]), (1) It is seen from (1) that the mean waiting time is a function of the first and second moments of the HOL file completion time. In Section III-A, we identify the optimal transmission strategies under the scheduling and coding modes of opera- tion, where optimality is in terms of minimizing the mean completion time. Then, in Section III-B, we provide closed form as well as asymptotic expressions for the first and second moments of the completion time under the identified optimal strategies. Numerical as well as asymptotic performance com- parison of the two transmission strategies will be provided in Section III-C. This investigation will reveal the delay gains of network coding with respect to traditional scheduling strategies in unreliable wireless systems.
A. Optimal Transmission Strategies
The aim of this section is to identify those coding and scheduling strategies that lead to minimum mean completion time of the HOL file, both in the presence and lack of CSI.
It can be seen by looking at (1) that the mean service time of a policy is the key factor in determining the maximum arrival rate λ that the policy can support with a finite delay. This is our motivation for focusing on minimizing this performance criterion. Next, we focus on the coding and scheduling cases separately.
4For other arrival processes, various bounds such as Kingman’s bound can be used to characterize the system delay.
1) Optimal Coding Strategy with and without CSI: It has been shown in the literature that linear coding is sufficient to achieve the maximum achievable rate for a single multicast session in general networks [12]. Noticing that the broadcast scenario is a special instance of a multicast transmission, we focus on linear coding strategies where the transmitted packet in slott is given by P[t] =
XK k=1
ak[t]Pk, with ak[t]∈ Fd for each k∈ {1, · · · , K}.
Proposition 1. The following randomized strategy is asymp- totically optimal as the field size d tends to infinity: The transmitter performs the following operation for the HOL file
RANDOMIZEDBROADCASTCODING(RBC):
While (File is incomplete)
Pick ak[t] uniformly at random from Fdfor each k;
Transmit P[t] =PK
k=1ak[t]Pk; t← t + 1;
Each receiver keeps the incoming packets that it could receive and then decodes all the packets{Pk}{k=1,··· ,K}as soon asK linearly independent combinations of the packets are collected (c.f. [8] and references therein). Finally, each receiver that successfully recovers the HOL file sends an acknowledgement to the transmitter.
Proof: The expected number of slots beforeK linearly independent combinations can be collected with Randomized Broadcast Coding (RBC) is given by PK
k=1[1− (1/d)k]−1. This expression can be upper-bounded byKd/(d− 1), which in turn can be made close to K even with reasonably low values ofd. Thus, for a large enough field size d, it is sufficient for each receiver to be active approximatelyK slots before it can decode the whole file. Notice that it is impossible to send the file with less thanK transmissions since at most one packet can be successfully transmitted in one transmission, and so RBC asymptotically (ind) achieves the best possible performance over all strategies.
Another important issue is the overhead related with this mode of transmission. Coding requires ⌈K log2d⌉ bits of overhead to contain the coefficients of the associated linear combination, whereas the packet size is⌈m log2d⌉ bits. Thus, form >> K, the overhead is negligible. Henceforth, we will consider this scenario, and ignore the overhead.
Notice that RBC is not only easy to implement, but also requires no knowledge of the channel state vector, and asymp- totically achieves the minimum mean completion time over all policies. We will see in Section III-A.2 that the optimal scheduling policy is much more difficult to characterize, even for symmetric channel conditions.
2) Scheduling Mode: In this mode, unlike in the coding mode, the presence or lack of CSI affects the performance.
Hence, these two cases will be studied separately.
a) Scheduling without CSI: In view of the assumptions that the transmitter receives feedback from each receiver only at the completion of the whole file and that the channels are symmetric, we can see that all packets in the HOL file have equal priorities. Therefore, we have the following result.
Proposition 2. Assuming NO-CSI and independent and iden-
tically distributed (i.i.d.) channels across time-slots and users, the optimal scheduling policy is Round Robin(RR), where Packet-k is transmitted in time-slots (mK + k) for m = 0, 1,· · · until all the receivers get the file.
Proof: This follows from the perfectly symmetric con- ditions assumed under this scenario.
b) Scheduling with CSI: Before we characterize the optimal scheduling rule with CSI, we demonstrate the subop- timality of scheduling compared to coding with the following key example.
Example 1: Consider the case of K = 3 and N = 3, i.e. three packets are to be broadcast to three receivers.
Consider the channel realizations C[1] = (0, 1, 1), C[2] = (1, 0, 1), C[3] = (1, 1, 0), and C[4] = (1, 1, 1). Thus, in the first four slots, each receiver can hear the transmission three times. The optimal scheduling rule would transmit P1, P2, P3
in the first three slots, leaving Receiver-i in demand for Packet- i in the fourth slot. Clearly, no scheduling rule can ever com- plete the file download at all three receivers in the fourth slot.
With coding, on the other hand, the following transmissions will complete the transmissions:(P1+ P2), (P2+ P3), (P1+ P3), (P1+ P2+ P3) (see Table I). It is not difficult to see that coding will never require more slots than is necessary for scheduling for all other realizations. Hence, we achieve strictly
better completion times with coding. ⋄
t = 1 t = 2 t = 3 t = 4
R1 − P2|(P2+P3) P3|(P1+P3) ?|(P1+P2+P3)
R2 P1|(P1+P2) − P3|(P1+P3) ?|(P1+P2+P3)
R3 P1|(P1+P2) P2|(P2+P3) − ?|(P1+P2+P3)
TABLE I. Demonstration of Example 1:Ricorresponds to Receiver-i,
‘−’ denotes OFF channel states, and the entrya|bgives the optimal transmissions with scheduling|coding, respectively. With scheduling, no choice of{Pi}in slot4can complete the file at all the receivers for the given channel realization.
OPTIMAL SCHEDULING RULE WITH CSI: We use Dy- namic Programming to find the characterization of the optimal scheduling policy. Given C[t], the scheduler can choose any one of the packets {P1,· · · , PK} for transmission. A little thought reveals the need for memory at the transmitter about the history of receptions of each receiver. For this purpose, we defineMi,k[t] to be the memory bit associated with Packet-k and Receiver-i. In particular, Mi,k[t] = 1 (or 0) implies that Receiver-i has not received (or has received) Packet-k in the slots 1,· · · , t − 1. Moreover, we will use M[t] to denote the matrix of memory bits[Mi,k[t]]k=1,··· ,Ki=1,··· ,N .
We let Π denote the set of feasible stationary policies that can be implemented by the transmitter. Each policy π ∈ Π defines a mapping from the pair (M[t], C[t]) to the set {1, · · · , K} describing the packet to be sent at time t. Note that the policy is stationary in the sense that it is only a function of the matrix and channel conditions at the time. The i.i.d. nature of the arrivals and departures imply that this is the optimal policy among all policies, including those that are time dependent.
To characterize the optimal policy we let Jπ(M, C) = E [# slots to reach θ with policy π | M[0] = M, C[0] = C] , where θ denotes the zero matrix. Then, J⋆(M, C) ,
minπ∈ΠJπ(M, C) is the minimum completion time of the optimal algorithm if it starts fromM and the first channel is C. Also, π⋆(M, C) , arg min
π∈Π Jπ(M, C) gives the optimal policy.
Observe that once we solve J⋆(M, C) for all C, we can compute J⋆(M) , EC[J⋆(M, C)] , where the expectation is over the channel realizations. Thus, J⋆(M) denotes the mean completion time of the optimal algorithm starting from M. Hence, we are interested in J⋆([1]N×K) where [a]N×K
denotes the all a matrix of dimensions N× K.
Before we write the recursion forJ⋆(M, C), let us define the functionf (·) where ˆM = f (M, C, k) implies that
Mˆi,k = Mi,k− Mi,kCi ∀i ∈ {1, · · · , N}, Mˆi,j = Mi,j ∀i ∈ {1, · · · , N}, j 6= k.
This function describes the next state of the memory matrix given that Packet-k is served and the channel matrix is C in the current slot. Then, we can write the following recursion:
J⋆(M, C) = arg min
k∈{1,··· ,K}
J⋆(f (M, C, k)) + 1{M6=θ}
,
where 1{A} is the indicator function of the eventA.
The monotone nature of the f (·) function enables us to compute J⋆(M, C) and π⋆(M, C) recursively starting from the base state J⋆(θ) = 0 (c.f. [4]). This DP formulation characterizes the optimal policy and its performance, and can be computed starting from a 1× 1 matrix and increasing N andK successively.
However, as N and K grows, the necessary number of operations required to find the optimal strategy grows expo- nentially and quickly becomes impossible to handle. Thus, we propose an efficient heuristic policy below and simulate its performance for comparison.
HEURISTICPOLICY: We have observed in the above dis- cussions that the optimal scheduling rule has a complicated structure. Yet, it is possible to find practical scheduling algo- rithms that performs close to the optimal. Here, we describe a heuristic policy that achieves near optimal performance based on numerical comparisons.
At any given time slott, let us denote the set of nodes with an ON channel (also called the set of active receivers) by A[t] , {i ∈ {1, · · · , N} : Ci[t] = 1}. Under the symmetric conditions that we assumed, the packet that would provide the most benefit should intuitively be transmitted over the channel.
We propose that the benefit of a packet be measured in the number of nodes inA[t] that has not yet received that packet.
The underlying idea is to transfer the maximum number of useful packets over the channel at any given time. These remarks point to the heuristic algorithm given next.
HEURISTICBROADCASTSCHEDULING(HBS):
If(t = 1)
Mi,k[t] ← 1 for all k ∈ {1, · · · , K}, i ∈ {1, · · · , N };
While
K
X
k=1 N
X
i=1
Mi,k[t] > 0
!
K[t], {k ∈ {1, · · · , K} : ∃i ∈ A[t] with Mi,k[t] = 1};
If(K[t] 6= ∅) T [t], arg max
k∈K[t]
X
i∈A[t]
Mi,k[t];
Pick a k⋆∈ T [t];
Mi,k⋆[t] ← 0 for all i ∈ A[t];
Transmit Packet-k⋆ over the channel at slot t;
t← t + 1;
In the algorithm, each packet inK[t] has at least one receiver with an ON channel in slot t which demands that packet.
Clearly, those packets that are not inK[t] should not be chosen for transmission. If K[t] 6= ∅, then we define T [t] to be the set of packets in K[t] that yield the most benefit in slot t.
Then, a packet from T [t] is picked for transmission in slot t. In our simulations, we considered a random picking of one of the packets in T [t]. However, the performance can be slightly improved by using more sophisticated methods.
For example, for N = 2, the packet picked from T [t]
may be chosen amongst those packets that has already been received by the OFF receiver. Then, every time a receiver is ON, it will receive a useful packet until all its packets are complete. Thus, this algorithm gives the optimal policy for N = 2. The generalization of the picking method to general K under asymmetric channel conditions is complicated and requires increasing memory to operate. On the other hand, the complexity of HBS at each iteration of the loop is O(KN ) and requires no extra memory, and hence it is relatively easy to implement.
B. Service Time Distributions
The goal of this section is to provide analytical and asymp- totic performance expressions for mean waiting time under the optimal coding and scheduling transmission strategies identi- fied in Section III-A. The exact analytical expressions provided here are in terms of infinite sums, and therefore do not yield much insight about the impact of system parameters on the performance. Here, we also derive asymptotic expressions to provide a sensitivity analysis with respect to key system parameters. We focus on the more realistic scenario of NO-CSI throughout this section. Our arguments are based on deriving expressions for the first and second moments of the completion time under coding and scheduling, and then using them in (1) to get the mean waiting time performances.
1) Performance Analysis of RBC: Let us define the random variable YiRBC as the number of slots before Receiver- i’s channel is ON K times, for i = 1,· · · , N. Then, the completion time under RBC for a given N and K, denoted by ZRBC(N, K), satisfies
ZRBC(N, K) = max
i∈{1,·,N }YiRBC, (2) which is the maximum of N Pascal variables of order K.
We will use mRBC1 and mRBC2 denote the first and second moments of ZRBC(N, K), respectively. Through algebraic
manipulations, we can derive closed-form expressions for these moments. As an example, the first moment is given by mRBC1 = K +
X∞ t=K
"
1− YN i=1
Xt τ=K
τ− 1 K− 1
qi(τ −K)pKi
!#
,
where
0 B B
@
n m
1 C C
A gives the number of sizem combinations of n elements, andqi, (1−pi). Similarly, a combinatorial expres- sion can be given for the second moment. For simplicity of exposition, we provide the second moment for the symmetric channel conditions:
mRBC2 = X∞
i=1
i2
Xi τ=K
τ− 1 K− 1
q(τ −K)pK
!N
− Xi−1 τ=K
τ− 1 K− 1
q(τ −K)pK
!N
. Although the exact expressions provided above can be used for numerical comparison, the expressions can be simplified by focusing on the asymptotic regime for the symmetric case.
Such an asymptotic study has the added advantage of revealing the gains of coding versus scheduling as a function of relevant system parameters. The asymptotic formulations are especially useful to understand the gains in dense networks, where an increasing number of transceivers are used within a fixed geographic area.
The next proposition, proved in [7], will be used in our subsequent analysis. It provides an expression for an infinite sum that is directly related to mRBC1 as will be noted in Proposition 4.
Proposition 3 ([7]). Letg(r) = βrαand letlq(·)be a short- hand for log1
q(·),then X
r≥0
1− (1 − g(r)qr)N
= lq N + α lq lq N + lq β +1 2 − γ
log q +h(lq N + α lq lq N + lq β) + o(1),
whereγis the Euler-Mascheroni constant (approximately equal to 0.5772), and h(·) is a periodic C∞-function of period 1 and mean value 0, whose Fourier coefficients are h(k) =ˆ
1 log qΓ
2ikπ logq
,fork∈ Z+.
The next proposition provides asymptotic expressions for mRBC1 andmRBC2 under symmetric conditions as a function ofN and K.
Proposition 4. Assume symmetric channel conditions, i.e., pi = pfor all i ∈ {1, · · · , N}, and letlq(·)be a shorthand for log1
q(·).Then, we have mRBC1 = lq T +1
2− γ
log q + h(lq T ) + o(1), mRBC2 = lq2T + lq T (1 + 2γ + 2g1(lq T )) +2
3
− γ
log q −(γ2+ (π2/6))
log2q + O((K− 1) lq lq N) +h(lq T ) + g2(lq T ) + o(1),
where T = N
p q
K−1
lq(K−1)N
(K−1)! , and h(·) is the periodic function of Proposition 3, andg1(·),andg2(·)are two periodic C∞-functions of period1and mean0.
Proof: We outline the proof of mRBC1 which is due to [7]. In the sequel, let us use Z and Yi as shorthands for ZRBC(N, K) and YiRBC(N, C) for convenience. Also, we useF⋆(·) generically to denote the cumulative distribution of the random variable ⋆ in the subscript.
Since {Yi} are i.i.d. Pascal random variables of order K, we have
FYc(m) := 1− FY(m) =
K−1X
k=0
m k
pkqm−k=: qmg(m),
where
g(m) :=
K−1X
k=0
m k
p q
k
∼ mK−1
(K− 1)!
p q
K−1
=: βmα, (3)
with β :=
p q
K−1 1
(K−1)!, and α := (K − 1). The last approximation is accurate for m ≫ K since the last term of the sum dominates, and
n k
∼ nk!k for n ≫ k. Then, we can write
mRBC1 = X
m≥0
(1− FmaxiYi(m))
= X
m≥0
1− (1 − g(m)qm)N .
Notice that the final expression is in the form of Proposition 3.
The proof is complete when we apply the result of Proposi- tion 3 withg(m) = βmα as defined above.
Next, we prove the expression for mRBC2 . Note that mRBC2 = E[Z2]
= X
k≥0
(1− FZ2(k))
= X
k≥0
1− FmaxiYi(⌊√ k⌋)
= X
k≥0
1−h
FY(⌊√
k⌋)iN
= X
k≥0
1−
1− g(⌊√
k⌋)qkN
, (4)
where⌊x⌋ is the largest integer that is less than or equal to x.
Note that following the arguments in (4), we can approximate (3) by replacingg(m) with βmαas argued above. In order to simplify the ⌊√
·⌋ in (4), we make a change of variable, and write
mRBC2 = X
r≥0
(2r + 1) 1− (1 − g(r)qr)N . (5)
We split the sum in (5) into two as
mRBC2 = E1+ 2E2, (6)
where
E1 , X
r≥0
1− (1 − g(r)qr)N ,
E2 , X
r≥0
r 1− (1 − g(r)qr)N
Notice that E1 = mRBC1 , which is already studied above.
Next, we derive a similar expression forE2. To that end, we define
E˜2 := X
r≥0
r (1− exp(−Nβrαqr)) Eˆ2 := X
r≥0
r (1− exp(−T qr)) ,
whereT := N β lqαN = N
p q
K−1
lq(K−1)N
(K−1)! . Then, we can write
E2 = (E2− ˜E2)
| {z }
∆˜
+ ( ˜E2− ˆE2)
| {z }
∆ˆ
+ ˆE2. (7)
We first derive an asymptotic expression for ˆE2 as a function of T. We then show that ˜∆ and ˆ∆ lead to negligible terms (asymptotically in terms ofN and K), and ˆE2dominates. Our derivation for ˆE2is based on taking its Mellin Transform, and then using Mellin inversion to find an explicit expression for its asymptotic form (see for example [17]). The Mellin transform, Eˆ2⋆(s) of ˆE2(T ) is given by
Eˆ2⋆(s) = Z ∞
0
Eˆ2(T )Ts−1dT = Γ(s) qs (qs− 1)2, forR(s)∈ (−1, 0), which uses the fact that
Z ∞ 0
(1− exp(−T ))Ts−1dT =−Γ(s), for R(s) ∈ (−1, 0).
Mellin inversion yields Eˆ2(T ) = 1
2πi
Z −12+i∞
−12−i∞ −Γ(s) qs
(qs− 1)2Ts−1ds.
Shifting the line of integration to the right gives the asymptotic behavior of ˆE2(T ) for T → ∞:
Eˆ2(T ) = log2q + 6 log2T − 12γ log(T ) − π2− 6γ2 12 log2q
+ X
l∈Z\{0}
Ress=2πi
log q(−Γ(s)) qs (qs− 1)2T−s
+ 1 2πi
Z M+i∞
M−i∞ −Γ(s) qs
(qs− 1)2Tsds, for any M > 0. The remaining integral is O(T−M) for any M > 0, and the sum of residues gives g1(lq T ) lq T +g2(lq T ) with two periodicC∞-functions of period 1 and mean value 0. Using these results together with our lq(·) notation, we can re-write the expression for ˆE2(T ) as
Eˆ2(T ) = lq2T
2 + (γ + g1(lq T )) lq T + 1 12
−(6γ2+ π2)
12 log2q + g2(lq T ) + o(1). (8)
Next, we study ˜∆ introduced in (7). We start by dividing the sum into two parts, while replacingg(r) by its approximation βrα, as follows:
∆˜ = X
r≤lq N
r
exp(−Nβrαqr)− (1 − βrαqr)N (9)
+ X
r>lq N
r
exp(−Nβrαqr)− (1 − βrαqr)N (10)
In the range when r≤ lq N, we have qr ≥ 1/N. Therefore, we get
(9)≤ X
r≤lq N
r
exp(−βrα)− (1 −βrα N )N
= X
r≤lq N
r exp(−βrα)
1− exp
−βrα 1
2N + o(1 N)
,
which follows from the fact that (1− Nx)N = exp(−x(1 +
1
2N + o(N1))). Next, using the approximation (1− e−y)≈ y for smally, which holds for large N, we can further simplify the previous sum to
X
r≤lq N
βr(α+1)exp(−βrα) 1
2N + o(1 N)
≤ β lq(α+2)N 1
2N + o(1 N)
,
which proves that as N tends to infinity, (9) tends to zero.
To find a bound on (10) we use the Taylor expansions for exN and(1− x)N to obtain
exN − (1 − x)N = O(N x2).
Substituting x = βrα in (10) yields
(10) = O
N β2 X
r>lq N
r2α+1q2r
= O (lq N)m N
,
for some positive integer m that can be computed using the poly-logarithmic function based on α. This shows that ˜∆ is negligible asymptotically asN → ∞.
Next, we focus on ˆ∆ that was introduced in (7). We begin by splitting the sum as
∆ =ˆ X
r≤lq N
re−N βrαqr
e−(T −N βrα)qr − 1 (11)
+ X
r>lq N
re−T qr
1− e−(N βrα−T )qr , (12)
whereT = N β lqαN. We study these sums separately. Note that (11)≤ 0, since in this range r ≤ lq N. Next, we focus on (12): Calculus shows that the summand of (12) is positive and increasing in the range wherer∈ (lq N, lq N +α lq lq N], and is upper-bounded byα lq lq N/ lqαN. Therefore we have the summation in this region which we further split into two
as follows: forr∈ (lq N, lq N + α lq lq N], we have X
lq N <r≤lq N +α lq lq N
re−N β lqαN qr
1− e−N β(rα−lqαN)qr
≤ O (α lq lq N)2 lqαN
, which tends to0 as N tends to infinity. Next we focus on the range whenr > lq N + α lq lq N. For notational convenience, we letx := r− lq N, and rewrite the sum in this range as
X
x>αlq lq N
h(x + lq N )e−β lqαN qx
×
1− e−β((x+lq N )α−lqαN)qxi
≤ X
x>αlq lq N
(x + lq N )
1− e−β((x+lq N )α−lqαN)qx
(a)≈ X
x>αlq lq N
(x + lq N ) lqαN
(1 + x
lq N)α− 1
(b)= β X
x>αlq lq N
(x + lq N ) lqαN Xα m=1
α m
xm lqmNqx
= βX
m=1
α m
lq(α−m)N X
x>αlq lq N
(xm+1+ lq N xm)qx
(c)= O
α lq(α−1)Nlq lq N (1 + lq N ) lqαN
= O(α lq lq N )
where the approximation(a) is due to the fact that (1−e−y)≈ y for small y; (b) follows from Binomial expansion; and (c) follows from the equality X
x>αlq lq N
xmqx= O m lq lq N lqαN
, and the fact that the dominant term occurs when m = 1.
Combining this result with (8) and the finding that ˜∆ is negligible, and then substituting these into (5) yields the expression formRBC2 stated in the the proposition.
Proposition 4 yields asymptotic formulations for the first and second moments of the maximum statistics of N Pascal distributed random variables of order K, and may, therefore, be of independent interest. Noting that we are primarily interested in understanding their effect on the mean waiting time, we next remark on the dominant terms. To that end, we study the dense network setting by fixing the file sizeK to a constant value and focusing on the asymptotic behavior asN increases. In this case,T behaves as lq N (1 + o(1/ lq N ))≈ lq N for large N. When this value is substituted in mRBC1,2 of Proposition 4, we can see that
mRBC1 ≈ lq N, and mRBC2 ≈ lq2N.
This is an interesting result when we note that mRBC2 ≥ mRBC1 2
, due to Jensen’s inequality. Thus, RBC asymptot- ically achieves the minimum possible second moment for the given first moment. Since we already know thatmRBC1 is the minimum achievable mean service time (cf. Proposition 1), this shows that RBC is also optimal in terms of minimizing the mean waiting time (cf. (1)).
2) Performance Analysis of RR: To compute the moments of the RR scheduler, we define Xki to be the number of transmissions of Pk before it is received by Receiver-i. Then,
Yi, max
k∈{1,··· ,K}
KXki+ k
gives the time slot when Receiver-i receives the whole file.
Finally, ZRR(N, K) , max
i∈{1,··· ,N }Yi gives the completion time of the RR scheduler. Similar to the RBC case, we use mRR1 and mRR2 to denote the first and second moments of ZRR(N, K), respectively. The next proposition provides tight bounds on mRR1 .
Proposition 5. Under symmetric channel conditions (i.e.pi= p∈ (0, 1)for alli), we have
mRR1 K = γ +
X∞ t=1
h1− 1 − qtKNi , for someγ∈ (1/2, 1).
Moreover, the asymptotic performance of the moments of the RR scheduler with respect toNfor fixedKsatisfies
K
2 + K lq(KN ) ≤ mRR1 ≤ K + K lq(KN), mRR2 ≥ (K/2 + K lq(KN))2.
Proof: The upper bound of 1 for γ is due to the fact thatk≤ K. The lower bound of 1/2 follows from stochastic coupling arguments and heavily relies on the symmetry of the channel distributions. In particular, consider a sample path of the channel state process, ω , (C[1], C[2], · · · ). We use i(ω) to denote the receiver that was the last to complete the file, and k(ω) to denote the index number of the last packet that Receiver-i(ω) received. With our earlier notation, Y (ω) gives the completion time of the file at Receiver-i(ω) under the given sample path. Also, notice that we have Y (ω) = Xk(ω)i(ω)(ω)K + k, for some integer Xk(ω)i(ω) that depends onω.
Next, for each sample path ω that leads to k(ω) ∈ {1, · · · , ⌊K/2⌋}, we will construct another sample path ˜ω that has the same probability of occurrence as ω, but leads toY (˜ω) = Xk(ω)i(ω)(ω)K + (K− k(ω)). This implies that
E[Y ]≥(K + 1)
2 + KE[max
i,k Xki]. (13) The construction of ω = ( ˜˜ C[1], ˜C[2],· · · ) follows the following rule:
C˜j[rK + l] =
Cj[rK + (K− l)], if r = Xk(ω)i(ω)(ω), j = i(ω),
l∈ {k(ω), K − k(ω)}, Cj[rK + l], otherwise.
It can be seen that under symmetric conditions this sample path has the properties listed above.
Next, we would like to find the second term in (13).
Due to i.i.d. assumptions,Xki are also i.i.d. with distribution P(Xki = m) = qm−1p, m = 1, 2,· · · . Since this distribution is independent of i and k, we can compute
E[max
i,k Xki] = X∞ t=1
h
1− 1 − qtKNi
. (14)
0 10 20 30 40 50 60
0 50 100 150 200 250 300 350 400
Number of users, N
Mean Completion Time
Performance of different Schemes for K=30, p=1/2
m1RR Upper bound m1RR Lower bound m1RBC
Heuristic Policy (HBS)
Fig. 2. Mean service time performance forK = 30andp = 1/2.
The first part of the proof is complete once (14) is substituted into (13).
To prove the asymptotic expressions, we note thatXki is a Pascal distributed random variable of order 1. Therefore the derivation of mRBC1 in Proposition 4 applies for computing E[max
i,k Xki] with N replaced with KN , and K replaced with 1. To obtain mRR2 we simply use Jensen’s inequality:mRR2 ≥
mRR1 2
.
C. Performance Comparison
In this section, we aim to demonstrate the coding gains on the mean waiting time for moderate and asymptotic values of N. To that end, we first provide numerical computations and simulations to compare the performance of various schemes we have discussed so far for moderate values of N and K.
A comparison of the first moments of RBC, RR and HBS is illustrated in Figure 2 as a function ofN , with K = 30, and each channel is ON or OFF equiprobably at every time slot.
The figure demonstrates the strength of the coding policy to the scheduling policy with and without CSI. We further observe that as N increases the advantage of using coding improves.
Figure 3 illustrates the waiting time performance of RBC versus RR for K = 30. It can be observed that the mean completion time gains are carried over to the mean waiting time performances. Notice that these huge gains are especially important to serve real-time traffic such as voice in unreliable networks.
Next, we provide the asymptotic gains of network coding compared to scheduling. We start by noting that for a fixed K, mRBC1 = lq N (1 + o(1/ lq N )) and mRBC2 = lq2N (1 + o(1/ lq2N )), whereas mRR1 = K lq N (1 + o(1/ lq N )) and
0 10 20 30 40 0
100 200 300 400 500 600
Number of receivers, N
Expected delay
E[W] for RBC E[W] for RR (lower bd) E[W] for RR (upper bd)
Fig. 3. Delay Performance of RBC versus RR forK = 30
mRR2 ≥ K2lq2N. Substituting these in (1) yields WRBC(λ) ∼ lq2N
2(1− λ lq N), λ < 1 lq N WRR(λ) ∼ K2lq2N
2(1− λK lq N), λ < 1 K lq N
We see that the maximum supportable arrival rateλ of RBC is K times that of RR. Moreover, when the system load is fixed to ρ∈ (0, 1) fraction of the available capacity, i.e., λRBC = ρ/ lq N and λRR= ρ/(K lq N ), then we have5
WRBC(λRBC) =WRR(λRR)
K2 .
In this section, we have seen that either with or without CSI, coding provides a considerable gain in the mean delay to download a given file to multiple receivers over a time-varying medium. Moreover, its operation is significantly easier than the scheduling policy. However, it requires an additional decoding operation at the receivers, which may or may not be critical depending on the file sizes and the computational capacity of the receivers.
IV. SERVINGMULTIPLEUNICASTSESSIONS
In this section, we consider the scenario whereN receivers with symmetric channel conditions demand unique flows, i.e.
F = N, and Nf = 1 for all f ∈ F. In this case, it is not clear whether coding will have the dominating behavior as it did in the broadcast scenario. Again, the availability of CSI is important. In Section IV-A, we will study some of the properties of the optimal scheduling and coding strategies.
Then, in Section IV-B, we will demonstrate the performance comparison through numerical computations.
A. Optimal Transmission Strategies
We will first study the scheduling case and then move on to the coding case.
5We note that the waiting time is measured per file, which consists ofK packets.
1) Scheduling for Multiple Unicasts: We again consider the case of CSI and no CSI.
a) Scheduling without CSI:: Without CSI, the obvious optimal scheduling is again Round Robin, except that it must be performed across files and across packets in each file. In particular, in the first round the first packet of each file is transmitted one after another, and in the next round the second packets are transmitted consecutively. When the end of a file is reached, we move to the first packet and continue until all the packets of a file is received by its receiver. Only then we remove that file from the RR scheduler and continue with the remaining ones.
In this scenario, we define the completion time as the amount of time required for all the HOL files to be completed at the interested receivers. As before, we assume that only after all the HOL files are transmitted, are the transmission of the next batch of HOL files are starts. This model can be extended to give different weights to different flows, and hence achieve different fairness distributions. The mean completion time performance of the above RR scheduling rule is easy to compute using recursive arguments, which is omitted here since it does not add any significant insights to our analysis.
b) Scheduling with CSI:: Here, the constraint is to serve at most one receiver at every time slot. This problem is a special case of a problem studied by Tassiulas and Ephremides in [21] with no arrivals to the system. The following policy is introduced in [21].
LONGESTCONNECTEDQUEUE(LCQ):
t← 0;
Qi← Ki for all i∈ {1, · · · , N};
Do
t← t + 1;
i⋆[t] ← arg max
1≤i≤N
{Ci[t]Qi};
if(Ci⋆[t] 6= 0)
TransmitPi⋆,Qi⋆; Qi⋆ ← max(0, Qi⋆− 1);
While
N
X
i=1
Qi>0
!
;
Return t; // Completion time
In the policy, Qi is used both as a pointer to the index of the next packet to be transmitted to Receiver-i, and also as the number of packets yet to be transmitted to Receiver-i.
Thus, LCQ is a myopic policy that favors the receiver with the maximum number of packets to be received among all connected receivers. We repeat the result of [21] for future reference.
Proposition 6 ([21]). Under symmetric channel conditions (i.e.pi = pfor alli), LCQ is minimizes the completion time over all scheduling policies. In other words,
ZLCQstZπ,
where TLCQ denotes the completion time under the LCQ policy andπis any other feasible scheduling policy6.
This result is very strong and implies that E[ZLCQ] ≤ E[Zπ] for any feasible scheduling policy π.
6stis a stochastic ordering as described in [21].
