[Paper Review] A Generalized Performance Evaluation Framework for Parallel Systems with Output Synchronization

Wasiur R. KhudaBukhsh, Sounak Kar+2 more|arXiv (Cornell University)|Dec 16, 2016

Cloud Computing and Resource Management23 references3 citations

TL;DR

This paper proposes a generalized Markov-additive process framework to evaluate performance in parallel systems with output synchronization, focusing on bounding tail probabilities of steady-state waiting times for both work-conserving and blocking servers. It derives computable bounds using martingale techniques and applies them to non-renewal arrivals, phase-type service behavior, and Markov-modulated systems, validated via simulations and real data calibration.

ABSTRACT

Frameworks, such as MapReduce and Hadoop are abundant nowadays. They seek to reap benefits of parallelization, albeit subject to a synchronization constraint at the output. Fork-Join (FJ) queuing models are used to analyze such systems. Arriving jobs are split into tasks each of which is mapped to exactly one server. A job leaves the system when all of its tasks are executed. As a metric of performance, we consider waiting times for both work-conserving and non-work conserving server systems under a mathematical set-up general enough to take into account possible phase-type behavior of the servers, and as suggested by recent evidences, bursty arrivals. To this end, we present a Markov-additive process framework for an FJ system and provide computable bounds on tail probabilities of steady-state waiting times, for both types of servers separately. We apply our results to three scenarios, namely, non-renewal (Markov-modulated) arrivals, servers showing phase-type behavior, and Markov-modulated arrivals and services. We compare our bounds against estimates obtained through simulations and also provide a theoretical conceptualization of provisions in FJ systems. Finally, we calibrate our model with real data traces, and illustrate how our bounds can be used to devise provisions.

Motivation & Objective

• To address the challenge of performance evaluation in parallel systems with output synchronization, where job completion depends on all tasks finishing.
• To model complex system behaviors such as bursty arrivals and dependent service times, which standard renewal and i.i.d. assumptions fail to capture.
• To develop a unified mathematical framework capable of handling non-renewal arrivals, phase-type service distributions, and Markov-modulated inputs and services.
• To provide computable upper bounds on the tail probabilities of steady-state waiting times, enabling performance provisioning and system design.
• To conceptualize and calibrate reactive and proactive provisions for dynamic load management in such systems.

Proposed method

• The authors model the system using a Markov-additive process (MAP) to capture state-dependent dynamics of arrivals and service times.
• They derive a martingale-based approach using the cumulant generating function and spectral properties of the transition kernel to bound tail probabilities.
• The framework leverages the Perron-Frobenius theorem for uncountable state spaces to ensure existence and positivity of the dominant eigenvalue and associated eigenfunction.
• Key components include the normalized right eigenfunction $ r(c, \theta) $, the cumulant function $ \zeta_k(s) $, and the essential supremum $ \phi(s) $ for bounding tail events.
• The method applies Doob’s maximal inequality to the martingale $ M_k(s) = \exp(sY_k - k\zeta(s))r(C_k,s) $, yielding exponential bounds on waiting time tail probabilities.
• The approach is calibrated to real data traces and applied to three scenarios: Markov-modulated arrivals, phase-type services, and joint Markov-modulated arrivals and services.

Experimental results

Research questions

• RQ1 How can a generalized performance evaluation framework be constructed for parallel systems with output synchronization under non-renewal and dependent arrival/service processes?
• RQ2 What are the computable upper bounds on the tail probabilities of steady-state waiting times in both work-conserving and blocking server models?
• RQ3 How can the Markov-additive process framework be specialized to model bursty arrivals and phase-type service behavior?
• RQ4 In what way can theoretical bounds be validated against empirical estimates from simulations?
• RQ5 How can provisions—both reactive and proactive—be conceptualized and applied within this framework for performance optimization?

Key findings

• The framework provides computable upper bounds on the tail probabilities of steady-state waiting times for both work-conserving and blocking servers, using martingale techniques and spectral analysis.
• For the case of i.i.d. exponential service times with rate $ \mu_i $, the MGF of the waiting time $ V $ is $ \mathrm{E}[e^{\theta V}] = \sum_{S \neq \emptyset} (-1)^{|S|+1} \frac{\sum_{i \in S} \mu_i}{\sum_{i \in S} \mu_i - \theta} $, and the mean waiting time is $ \mathrm{E}[V] = \sum_{S \neq \emptyset} (-1)^{|S|+1} \frac{1}{\sum_{i \in S} \mu_i} $.
• The theoretical bounds on tail probabilities closely match empirical complementary cumulative distribution functions (CCDFs) obtained from simulations across all three application scenarios.
• The model is successfully calibrated to real data traces, demonstrating its practical relevance for system provisioning.
• The framework enables the conceptualization of reactive and proactive provisions, with a simple reactive provision illustrated for illustrative purposes.
• The derived bounds are shown to be effective for performance evaluation and can guide system design under complex, realistic traffic and service patterns.