Numerical Performance Evaluation of Heterogeneous Multi-server Models with Breakdowns and FCFS, LCFS-PR, LCFS-NPR Repair Strategies Abhilash PG Scholar Department of CSE National Institute of Technical Teachers Training & Research Chandigarh, 160019 Email: abhilashsharma@gmail.com Ram Chakka Director (Research & Development) Meerut Institute of Engineering & Technology Meerut, 250005 Email: ramchakka@yahoo.com Rama Krishna Challa Associate Professor Department of CSE National Institute of Technical Teachers Training & Research Chandigarh, 160019 Email: ramakrishna.challa@gmail.com Abstract—Multi-server performance models are used in the modeling of advanced computing systems, communication nodes and networks, and manufacturing systems. In [15], [5], a fast method known as Spectral Expansion was developed. Extension of that method to finite capacity multi-server systems was done in [16]. Application of this model to networks was done in [19]. Extension to heterogeneous systems with breakdowns and repairs was done in [18] in which only a few repair strategies were considered. Principles towards extending the range of repair strategies was attempted in [20], but those models were neither evaluated numerically nor compared. The purpose of the present paper is to show how heterogeneous multi-server systems with FCFS, LCFS (PR), LCFS (NPR) repair strategies can be evaluated numerically using the Spectral Expansion algorithm. Index Terms—Multi-server systems, heterogeneous machine repairman problem, time dependent failures, Markov processes. I. I NTRODUCTION To model multiprocessor systems [9], [14], nodes in com- munication networks[19] and flexible machine shops [6], [8], [7], [21] in manufacturing environment, multi-server system models are often used. Also, to take into account the effect of server breakdowns and repairs [12], [3], [4], [17], [13], [10], [7], [20], much efforts have been put up in developing the models. However, the progress is considerably small in mod- elling heterogeneous multi-server systems with breakdowns [18], [2], [21]. A heterogeneous multi-server system with unbounded queuing capacity, independent server breakdowns and a single repair facility was considered in [18] where exact numerical solution to performability of such a system serving a Poisson system of jobs was presented. But there, a limited number of repair strategies, such as servers with given PR-priority, non preemptive priority for repair, and round- robin repair were considered. In [2], [21], some interesting theoretical extensions to [18] were carried out. However, due to the lack of straight forward modelling approaches, important repair strategies such as FCFS, LCFS etc. were not modelled. In this paper, the approaches have been developed to model complicated repair strategies such as FCFS and LCFS. The repair strategies, corresponding to finite and infinite queue capacity, are compared systematically. For LCFS repair strat- egy, the case of PR (Preemptive Resume) and NPR (Non- Preemptive Resume) are considered. The paper is organized as follows. In section-2, hetero- geneous multi-server systems are discussed with breakdowns and repair strategies. Here, the system is modelled as a QBD process. Three alternate repair strategies are described and modelled in section-3. The corresponding numerical results are presented in Section-4 and in section-5 conclusions are persented. II. MULTI - SERVERS WITH NON-I DENTICAL SERVERS The heterogeneous multi-server system is shown in Fig. 1. It consists of  non-identical parallel servers, numbered 1, 2, ⋅⋅⋅ ,, with a common queue. Here servers are assumed to be non-identical because of their different service rates. Including the jobs in service, the queue is of capacity , ( ≤  ≤ ∞) and can be bounded (finite length) or unbounded (infinite length). Job joins the queue if there is queuing space else it gets discarded. In addition, it is assumed that, all jobs are homo- geneous and arriving into the system at a rate  as a Poisson process. Thus, the service times of the jobs serviced by  ℎ server, where ( =1, 2, ⋅⋅⋅ ), will be exponentially dis- tributed with mean 1/  . Server, say  ℎ , executes jobs only during its operative periods. During an operative period, the server is capable of its intended operation, whether working or idle. Operative periods are exponentially distributed with mean 1/  which is equivalent to a constant failure rate of   . When server  breaks down, repair time is exponentially distributed with mean 1/  . Let there be  repairmen ( ≤ ), each 567 978-1-4673-4529-3/12/$31.00 c 2012 IEEE