Notation and Terminology N ( t ) = # of customers in the system at time t ³ 0 P k ( t ) = probability exactly k customers in system at time t , given # in system at time 0 s = # of parallel servers λ k = mean arrival rate (expected # of arrivals per unit time) μ k = mean service rate (expected # of departures per unit time) (Both λ k and μ k assume k customers are in system) If λ n does not depend on # of customers in system, λ n = λ . If there are s servers, each with the same service rate, then μ n = s μfor n >= s and μ n = n μ for 0 <= n < s . s μ= customer service capacity per unit time ρ= λ/ s μ= utilization factor (traffic intensity) The systems we study will have ρ < 1 because otherwise the # of customers in the system will grow without bound. We will be interested in the steady-state behavior of queuing systems (the behavior for t large). Obtaining analytical results for N ( t ), P n ( t ), . . . for arbitrary values of t (the transient behavior) is much more difficult. Notation for steady-state analysis π n = probability of having exactly
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