1. For all s ∈ Dρ,exp ρ(s) is the simple maximal eigenvalue of ˜L and the corresponding right eigenfunction{˜r(c, s); c∈E}satisfying
exp ρ(s) ˜r(c, s) =
Z
R˜L(c, dτ; s)˜r(τ, s),
is positive and bounded above.
2. Tail probabilities of the steady-state waiting time defined in (4.6.1) are bounded above by P(W˜ ≥w)≤ϕ(θ)exp −θw, (4.6.2) where θ := sup{s > 0 | ρ(s) ≤ 0}and ϕ(s) := ess sup{1(Z1 > 0)/˜r(C1, s)}after having normalised r(., θ)so that E[˜r(C0, θ)] =1.
Example 4.6.1. Suppose the arrival process is renewal, i.e.,E = {1}. Then, instead of solving an eigenvalue problem, we solve a nonlinear equation involving the MGF and the Laplace transform of the service times and the inter-arrival times. Suppose the ser- vice times of the n-th server are exponentially distributed with rate µn, for n∈ [N]. Also assume the inter-arrival times are exponentially distributed with parameter λ. Since the minimum of a finite collection of exponential random variables is itself exponen- tially distributed, the decay rate θ in Corollary4.6.1 is found by solving the following equation ∑n∈[N]µn ∑n∈[N]µn−θ ! λ λ−θ =1,
which yields a closed-form solution θ = ∑n∈[N]µn−λ. The upper bound on the tail probabilities is then found by plugging in this value of θ in (4.6.2).
Interestingly, we would have obtained the same decay rate if we had one single server with combined capacity, i.e., whose service times were exponentially distributed with rate ∑n∈[N]µn. Therefore, as far as the asymptotic decay rate of the tail probabilities of the steady-state waiting times is concerned, an FJ system with N work-conserving servers governed by a purging replication strategy has the same performance as a queue- ing system with one single server whose service rate is equal to the total of the individual service rates. It is remarkable that even a simple bound such as the one obtained in this example can present such insights into the performance of nontrivial FJ systems with replication strategies.
4.6.3 Renewal Processes as a special case
Several previously known results on Fork-Join systems where a renewal arrival process was assumed (e.g., the renewal cases in KhudaBukhsh, Rizk, et al. (2017) and Rizk,
Poloczek, and Ciucu (2015), and also the FJ system in Chapter 3) can be retrieved by settingE = {1}. In this case, following Algorithm 4.1 and Algorithm4.2, the bounds turn out to be
P(W≥w)≤
∑
n∈[N]
exp −θnw, and P(W′≥w)≤exp −θw, (4.6.3) where
θn= sup{s>0|E[exp sSn,1]E[exp −sA1
]≤1}, θ= sup{s>0|E[exp s max
n∈[N]Sn,1
]E[exp −sA1
]≤1}. This further enhances the applicability of our results.
In the next chapter, we shall apply the results obtained for general FJ systems in the previous and the current chapter to the collaborative uploading problem discussed in Section1.1. In particular, we shall use the bounds to devise uploading strategies for this scenario.
5
C O L L A B O R AT I V E U P L O A D I N G
In this chapter, we consider the collaborative uploading problem described in Section1.2. Our goal is to find optimal collaborative uploading strategies. We differentiate between the intermittent (devices such as sensors sending data on a coarse time scale) and the continuous collaborative uploading (devices continuously streaming video footage, e.g., using Facebook Live (Facebook Live 2018), Perisope (Twitter, Inc 2018) ) cases and study
them separately.
We analyse replication and allocation strategies for the collaborative uploading sce- nario. For the continuous stream uploading case, we use an FJ queueing system for- mulation that captures the ability to split data into chunks that are transmitted over multiple paths, and finally merged when all chunks are received. The results developed in Chapters 3 and 4 are utilised to design optimal strategies for the stream uploading case. We provide closed-form expressions for the mean upload latency in the inter- mittent uploading case, allowing a comparison between a replication and an allocation (splitting) strategy. We find optimal strategies for given path latencies. In doing so, we also show numerical results suggesting near-optimality of the proportional allocation.
5.1 m o d e l l i n g a p p r oa c h
Here, we present an overview of our approach, which consists of (i) defining an ap- propriate performance metric, and (ii) framing an appropriate optimisation problem thereafter.
t h e i n t e r m i t t e n t c a s e We characterise the intermittent case as one where the time intervals between two successive uploads are so large that there is no self-induced queueing. Then, aspects such as cross-traffic can be described by means of the statistical properties of the path latencies alone. A primary device uploading data intermittently aims to minimise the upload latency, i.e., the time until the data reaches the cloud. Given multiple paths, the primary device may split the data into chunks that are transmitted or replicated over the available paths. The upload latency being a stochastic quantity, it is natural to consider its mean as a performance metric and optimise it over all possible splitting/replication configurations. In Section5.2, we express the upload latency as an order statistic of the individual upload times over the different paths, making the theory of order statistics a useful tool in our analysis.
t h e s t r e a m u p l oa d i n g c a s e In the case of continuous upload of a data stream, e.g., a primary device uploading a live video to the cloud, there is a notion of waiting before each data chunk can be uploaded and hence, that of queueing. We call the event of new data generation and passing by the application to the lower layers on the primary device, an arrival of a new data batch. Each data batch is split into chunks of various sizes that are transported over several paths. Paths are characterised by a random service
time required to transport the assigned chunks. Finally, the data batch reaches the cloud when all of its chunks are received. Therefore, such systems are naturally modelled as FJ queueing systems.
5.2 i n t e r m i t t e n t c o l l a b o r at i v e u p l oa d i n g
In the following we consider the intermittent uploading case of data of size K over N possibly heterogeneous paths (e.g., sensor or monitoring devices uploading data on a coarse time scale). Assume that the data can be divided into N smaller chunks consisting of packets. Then, every k= (k1, k2, . . . , kN)∈Λ(N, K)is a valid allocation vector, where ki denotes the number of packets to be sent via path i and Λ(N, K)denotes the set of all non-negative integer solutions to the Diophantine equation∑N
i=1ki = K, for N, K ∈
N. We denote the random amount of time taken to transport the j-th packet out of
the ki packets allocated to path i by Di,j. Here, Di,j may capture different phenomena that impact the transmission time over a path, such as resource allocation, transmission collisions, and retransmissions. Assume that for each i∈ [N], the random variables Di,j’s are mutually independently distributed1
. Recall that the data consisting of K packets can be reconstructed only after all the packets have arrived. Therefore, the upload latency can be expressed as D :=max(D(k1)
1 , D (k2) 2 , . . . , D (kN) N )where D (ki) i :=∑ ki j=1Di,jfor ki>0 denotes the amount of time taken by path i to transport ki packets, and by convention, D(0)i :=0∀i ∈ [N]. The random variable D measures the total amount of time taken to transport all the packets over N different paths. In this work, we consider
ψ(k):=E[D] =E[max(D1(k1), D(k2N), . . . , D(kNN))],
the expected upload time given an allocation k, as our performance metric. The density function of D(ki)
i is given by the ki-fold self-convolution of the density function of Di,j due to independence. Let us denote the Cumulative Distribution Function (CDF) of D(ki)
i by F (ki)
i . Stacking into a column vector F(k):= (F (k1) 1 , F (k2) 2 , . . . , F (kN) N )T, we express the expected values of the order statistics of D(k1)
1 , D (k2)
2 , . . . , D (kN)
N as an operator µ on F(k) (see RemarkC.1.1in Appendix C.1.1). Since ψ(k)is the first moment of the N-th order statistic, we get
ψ(k) =µNF(k)=
∑
j∈[N](−1)j+1MjF(k), (5.2.1)
where µN andMj are operators defined in Appendix C.1.1. The optimal allocation is found by minimising ψ, i.e.,
kopt:= argmin k∈Λ(N,K)
ψ(k). (5.2.2)
Note that when the path characteristics are unknown, we can perform statistical infer- ence. In the following, we show some illustrative examples with computable koptbefore generalising this allocation scheme to include replication strategies.
1 Mutual independence, although not necessary for the subsequent analysis, is assumed for the sake of sim- plicity. In order to account for possible dependencies observed in real-world applications, one needs to addi-