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network of single-server queues with the following characteristics:

  • M = # of queues in the system, not counting queue 0 which represents the outside world

  • \mu_i = service rate at queue i

  • \lambda_i = total rate at which jobs arrive at queue j

  • orall i,1\leq i\leq M: ho_i = utilization of the service at queue i = rac {\lambda_i}{\mu_i} < 1

  • n_i(t) =# of jobs in queue i at time t

  • n(t)=(n_1(t), n_2(t), ..., n_M(t))^T= the system state at time t

  • P(k_1, k_2, ..., k_M, t)=Pr(n(t)=k_1, k_2, ..., k_M)^T)

  • P(k_1, k_2, ..., k_M)=\lim_{t o\infty}P(k_1,k_2,...,k_M,t)

  • Arrivals from the outside world are Poisson. All queues have exponential service time distributions.



PRODUCTION FORM OF JACKSON'S NETWORK

:P(k_1,k_2,...,k_M)=\prod_{i=1 o M}[( rac{\lambda_i}{\mu_i})^{k_i}(1- rac{\lambda_i}{\mu_i})]=\prod_{i=1 o M}[(1- ho_i) ho_i^{k_i}]

(where ho_i= rac{\lambda_i}{\mu_i})


SEE ALSO



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