Stochastic bounds in Fork–Join queueing systems under full and partial mapping

Abstract

In a Fork–Join (FJ) queueing system, an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks’ response times. This queueing system is useful to the modeling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze. This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems under full (bijective) and partial (injective) mapping of tasks to servers. We consider four practical scenarios by combining (1a) renewal and (1b) non-renewal arrivals, and (2a) non-blocking and (2b) blocking servers. In the case of non-blocking servers, we prove that delays scale as $\mathcal {O}(\log N)$, a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of $\log N$ dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing “makes sense” from a queueing perspective for two paths only, i.e., response times drop the most when $N=2$; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions.

Appendix

We assume throughout the paper that all probabilistic objects are defined on a common filtered probability space $\left( \Omega ,\mathcal {A},\left( \mathcal {F}_n\right) _n,\mathsf {P}\right)$. All processes $\left( X_n\right) _n$ are assumed to be adapted, i.e., for each $n\ge 0$, the random variable $X_n$ is $\mathcal {F}_n$-measurable.

Definition 1

(Martingale) An integrable process $\left( X_n\right) _n$ is a martingale if and only if for each $n\ge 1$

$$\begin{aligned} \mathsf {E}\left[ {X_{n}\mid \mathcal {F}_{n-1}}\right] =X_{n-1}. \end{aligned}$$

(44)

Further, X is said to be a sub-(super-)martingale if in (44) we have $\ge$ ($\le$) instead of equality.

The key property of (sub, super)-martingales that we use in this paper is described by the following lemma:

Lemma 1

(Optional Sampling Theorem) Let $\left( X_n\right) _n$ be a martingale, and K a bounded stopping time, i.e., $K\le n$ a.s. for some $n\ge 0$ and $\{K=k\}\in \mathcal {F}_k$ for all $k\le n$. Then

$$\begin{aligned} \mathsf {E}\left[ {X_0}\right] =\mathsf {E}\left[ {X_K}\right] =\mathsf {E}\left[ {X_n}\right] . \end{aligned}$$

(45)

If X is a sub-(super)-martingale, the equality sign in (45) is replaced by $\le$ ($\ge$).

Proof

See, for example, [7]. $\square$

Note that for any (possibly unbounded) stopping time K, the stopping time $K\wedge n$ is always bounded. We use Lemma 1 with the stopping times $K\wedge n$ in the proofs of Theorems 1, 2, 3 and 4.

Lemma 2

Let $c_k$ be the Markov chain from Fig. 4 and K be the stopping time from (11). Then the distribution of $(c_K\mid K<\infty )$ is stochastically smaller than the steady-state distribution of $c_k$, i.e.,

$$\begin{aligned} \mathsf {P}\left[ {c_K=2\mid K<\infty }\right] \le \mathsf {P}\left[ {c_1=2}\right] , \end{aligned}$$

or, equivalently,

$$\begin{aligned} \mathsf {E}\left[ {h(c_K)}\mid {K<\infty }\right] \ge \mathsf {E}\left[ {h(c_k)}\right] , \end{aligned}$$

for all monotonically decreasing functions h on $\{1,2\}$.

Proof

Using Bayes’ rule and the stationarity of the process $c_k$, we have:

$$\begin{aligned} \mathsf {P}\left[ {c_K=2\mid K<\infty }\right]&=\sum _{k=1}^{\infty }\mathsf {P}\left[ {c_k=2\mid K=k}\right] \mathsf {P}\left[ {K=k}\right] \\&= \sum _{k=1}^{\infty }\mathsf {P}\left[ {K=k\mid c_k=2}\right] \mathsf {P}\left[ {c_k=2}\right] \\&= \mathsf {P}\left[ {c_1=2}\right] \sum _{k=1}^{\infty }\mathsf {P}\left[ {K=k\mid c_k=2}\right] . \end{aligned}$$

Since $L_1$ is stochastically smaller than $L_2$, we have for any $k\ge 1$

$$\begin{aligned} \mathbb {P}&[K=k\mid c_k=2]\\&=\mathsf {P}\left[ {t_k\le \max _n\sum _{i=1}^k x_{n,i}-\sum _{i=1}^{k-1}t_i-\sigma , \max _n\sum _{i=1}^{k-1}(x_{n,i}-t_i)<\sigma \mathrel {\bigg |} c_k=2}\right] \\&\le \mathsf {P}\left[ {t_k\le \max _n\sum _{i=1}^kx_{n,i} -\sum _{i=1}^{k-1}t_i-\sigma , \max _n\sum _{i=1}^{k-1}(x_{n,i}-t_i)<\sigma }\right] \\&=\mathsf {P}\left[ {K=k}\right] . \end{aligned}$$

Hence $\sum _{k=1}^{\infty }\mathsf {P}\left[ {K=k\mid c_k=2}\right] \le 1$, which completes the proof. $\square$