# Combined performance and availability analysis of distributed resources in grid computing

## Abstract

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.

### Keywords

Grid computing Performance Availability Markov reward model Mean sojourn time### Abbreviations

- FTCS
Fault tolerant computer system

- CTMC
Continuous time Markov chain

- MRM
Markov reward model

- RMS
Resource management system

- GSPN
Generalized stochastic Petri net

- SAN
Stochastic activity network

### List of symbols

- \(t\)
Time

- \(i\)
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

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

- \(\Omega \)
State space of CTMC

- \(N\)
Number of states in CTMC

- \(Q\)
Generator matrix

- \(P(t)\)
Transient probability vector

- \(p_{0}\)
Initial probability vector

- \(\pi \)
Steady state probability vector

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

- \(X(t)\)
Instantaneous reward rate of the related MRM

- \(r\)
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

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