Numerical Solution to the Performability of a Multiprocessor System with Reconfiguration and Rebooting Delays Orhan Gemikonakli Tien Van Do Middlesex University Dept. of Telecommunications White Hart Lane London N17 8HR Budapest University of Technology and Economics United Kingdom Hungary E-mail: o.gemikonakli@mdx.ac.uk E-mail: do@hit.bme.hu Ram Chakka Enver Ever RGMCET Middlesex University Nandyal White Hart Lane India 518501 London N17 8HR, UK E-mail: ramchakka@yahoo.com E-mail: e.ever@mdx.ac.uk KEYWORDS Queuing, multi-processor systems, rebooting and reconfiguration delays. ABSTRACT Multiprocessor system models are extensively used in modelling transaction processing systems, nodes in communication networks, and flexible machine shops with groups of machines. Such systems clearly, are prone to break-downs. Even if cover is provided with some probability c, there will be rebooting and/or reconfiguration delays to resume operation following the break-down of a processor. In this paper, the performance modelling of a multiprocessor system, with identical processors, serving a stream of arriving jobs is considered. To account for delays due to reconfiguration and rebooting, such systems are modelled and solved for exact performability measures for both bounded and unbounded queuing systems. INTRODUCTION Multiserver system models are useful to model multiprocessor systems (Trivedi 2002; Harrison and Patel 1993), nodes in communication networks, and flexible machine shops (Stecke and Kim 1989; Stecke 1992; Righter 1996; Buzacott and Shantikumar 1993; Fiems et al. 2004) in a manufacturing environment. In this paper we develop approaches to model homogeneous multiprocessor systems with reconfiguration and rebooting delays by suitably extending the resulting quasi birth death (QBD) process in the performance models of multiprocessor systems with breakdowns and repair strategies (Chakka and Mitrani 1992; Chakka et al. 2002). This was considered in (Trivedi and Sathaye 1990) and an approximate performance model based on Markov reward models was presented. In this paper, we derive an exact solution for the steady state probabilities of the same problem using the spectral expansion method. The effects of reconfiguration and rebooting delays are analysed. The paper is organised as follows. The next section presents the homogeneous multiprocessor system with breakdowns and repairs considered in this work, and models the system as a QBD process. The section on modelling reconfiguration and rebooting delays in multiprocessor systems deals with a homogeneous multiprocessor system with breakdowns, repairs, and with reconfiguration and rebooting delays (Trivedi and Sathaye 1990). Exact solution for steady state performability for is derived using the spectral expansion method in the section on steady state solution. The model considered is very useful in the computer industry. Exact solution to this model and numerical results are also presented for both unbounded and bounded systems. MULTIPROCESSOR SYSTEM WITH IDENTICAL PROCESSORS The homogeneous multiprocessor system, shown in Figure 1, consists of K identical parallel processors, numbered 1, 2, …, K, with a common queue. The queue is of capacity L (finite or infinite L K), including the jobs in service. Jobs arrive into the system in a Poisson stream at rate , and join the queue. Jobs are homogeneous and the service rates of the processors assumed identical. Thus, the service times of jobs serviced by processor k (k=1, 2, …, K) are distributed exponentially with mean 1/ . However, processor k executes jobs only during its operative periods (during an operative period the processor is capable of its intended operation, whether working or idle), which are distributed exponentially with mean 1/ (equivalent to a constant failure rate of when operative). At the end of an operative period, processor k breaks down and requires an exponentially distributed repair time with mean 1/ . The number of repairs that may proceed in parallel could be restricted. This is expressed by saying that there are R repairmen (R K), each of whom can work on at most one repair at a time. Thus, an inoperative period of a processor would also include the