The Journal of Supercomputing

, Volume 69, Issue 2, pp 827–844 | Cite as

Combined performance and availability analysis of distributed resources in grid computing

  • Reza Entezari-Maleki
  • Ali Mohammadkhan
  • Heon Young Yeom
  • Ali Movaghar


This paper presents a mathematical model to evaluate the performance of grid resources when availability of the resources is taken into account. The proposed model uses continuous time Markov chains (CTMCs) to model the failure-repair behavior of a grid resource. In grid computing environment, a resource not only may fail during task execution, but also it can cancel its membership at any time. Hence, the proposed CTMC considers the availability of a grid resource to a grid user in both failure and membership refusal situations. After modeling the availability of the resource, the mean sojourn time of grid tasks in each of the availability states is estimated. Assigning the mean sojourn times of the tasks as performance levels to each of the CTMC’s states, a Markov reward model (MRM) representing the combined performance and availability measures is obtained. Computing the cumulative state probability of the CTMC and multiplying reward rates of the MRM’s states to each of the corresponding state probabilities, the expected accumulated sojourn time of grid tasks in each of the grid resources is achieved. An illustrative example is presented and the results obtained from the proposed model are reported in cases where various scheduling disciplines are considered inside the grid resource to simultaneously service grid and local tasks.


Grid computing Performance Availability Markov reward model Mean sojourn time 



Fault tolerant computer system


Continuous time Markov chain


Markov reward model


Resource management system


Generalized stochastic Petri net


Stochastic activity network

List of symbols




Index of the system

\(\alpha \)

Failure rate of the resource

\(\beta \)

Repair rate of the resource

\(\lambda _\mathrm{l}\)

Local tasks arrival rate

\(\mu _\mathrm{l}\)

Resource service rate for local tasks

\(\lambda _\mathrm{g}\)

Grid tasks arrival rate

\(\mu _\mathrm{g}\)

Resource service rate for grid tasks

\(\mu \)

Total service rate of the resource


\(\hbox {t}\ge 0\), a random process representing the related CTMC

\(\Omega \)

State space of CTMC


Number of states in CTMC


Generator matrix


Transient probability vector


Initial probability vector

\(\pi \)

Steady state probability vector


Cumulative state probability vector during time period \(\left[ {0,t} \right) \)


Instantaneous reward rate of the related MRM


Reward rate vector over \(Z( t)\)

\(\Phi (t)\)

Accumulative reward over the period \(\left[ {0,t} \right) \)

\({E\left[ {X(t)}\right] }\)

Expected instantaneous reward rate

\({E\left[ X \right] }\)

Expected steady state reward

\({E\left[ {\Phi (t)} \right] }\)

Expected accumulated reward rate



The authors would like to thank Iran Telecommunication Research Center (ITRC) for their support. This research was also supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the CPRC (Communications Policy Research Center) support program supervised by the KCA (Korea Communications Agency)(KCA-1194100004).


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Reza Entezari-Maleki
    • 1
  • Ali Mohammadkhan
    • 1
  • Heon Young Yeom
    • 2
  • Ali Movaghar
    • 1
  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran
  2. 2.School of Computer Science and EngineeringSeoul National UniversitySeoulSouth Korea

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