An M/M/2 parallel system model with pure space sharing among rigid jobs

We analyze a parallel system with two identical servers and pure space sharing among rigid jobs. The parallel system is modelled as an M/M/2 queue with two types of jobs. Jobs of one type are parallel ones and require two servers while jobs of the other type need one server to start execution. Analysis of the system leads to a quartic polynomial for which a real root inside the unit disk is located. This root is then used to derive closed-form expressions for the mean queue length, mean response time as well as mean utilization of the system. The maximal utilization of the servers is also found. In addition, we provide an approximation that simplifies the previously found exact but complex expressions and gives insight into the impact of different parameters on system performance. Numerical experiments are presented and our formulae are validated by simulation.

[1]  S. Neumark.  Solution of Cubic and Quartic Equations , 1965 .

[2]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[3]  Dave Auckly.  Solving the Quartic with a Pencil , 2007, Am. Math. Mon..

[4]  Alma Riska,et al.  Aggregate matrix-analytic techniques and their applications , 2002 .

[5]  Anca I. D. Bucur,et al.  The Performance of Processor Co-Allocation in Multicluster Systems , 2003, CCGRID.

[6]  H. R. Gail,et al.  Spectral analysis of M/G/1 and G/M/1 type Markov chains , 1996, Advances in Applied Probability.

[7]  Peter G. Harrison,et al.  Performance modelling of communication networks and computer architectures , 1992, International computer science series.

[8]  Helen D. Karatza.  Job scheduling in heterogeneous distributed systems , 2001, J. Syst. Softw..

[9]  Vincenzo Grassi,et al.  An Analytical Model for a Parallel Fault-Tolerant Computing System , 1999, Perform. Evaluation.

[10]  Giuseppe Serazzi,et al.  Processor Saving Scheduling Policies for Multiprocessor Systems , 1998, IEEE Trans. Computers.

[11]  Onno Boxma,et al.  Queueing-theoretic solution methods for models of parallel and distributed systems , 1994 .

[12]  Thomas G. Robertazzi,et al.  Computer Networks and Systems , 1994, Springer US.

[13]  Guy Latouche,et al.  A matrix-analytic approach to fluid queues with feedback control , 2005 .

[14]  Marios C. Papaefthymiou,et al.  Stochastic Analysis of Gang Scheduling in Parallel and Distributed Systems , 1996, Perform. Evaluation.

[15]  Kenneth C. Sevcik,et al.  Implementing Multiprocessor Scheduling Disciplines , 1997, JSSPP.

[16]  Raymond M. Bryant.  Maximum Processing Rates of Memory Bound Systems , 1982, JACM.

[17]  Kento Aida,et al.  Job Scheduling Scheme for Pure Space Sharing Among Rigid Jobs , 1998, JSSPP.

[18]  Henryk Krawczyk,et al.  An Approximation Algorithm for Diagnostic Test Scheduling in Multicomputer Systems , 1985, IEEE Transactions on Computers.

[19]  Wei Sun,et al.  ADAPTLOAD: effective balancing in clustered web servers under transient load conditions , 2002, Proceedings 22nd International Conference on Distributed Computing Systems.