Birth-Death Process Birth – arrival of a customer to the system

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Presentation transcript:

Birth-Death Process Birth – arrival of a customer to the system Death – departure of a customer from the system N(t) – random variable associated with the state of the system at time t (i.e. the number of customers, n, in the system at time t). Assumptions – customers inter-arrival times are exponential at a rate of ln and their service times are exponential at a rate of mn Queuing System Departures - mn Arrivals - ln

Rate (Transition) Diagram for Birth-Death Process l l l l l 1 2 n-2 n-1 n … … m m m States of the system (e.g. number of customers in the system) Transition rates (e.g. arrival rates and service rates)

… … M/M/1 System l l l l l 1 2 n-2 n-1 n m m m m m 1 2 n-2 n-1 n … … m m m m m Balance Equations: Node 0) lp0 = mp1 Node 1) lp0 + mp2 = lp1 + mp1 Node 2) lp1 + mp3 = lp2 + mp2 : : : : Node n) lpn-1 + mpn+1 = lpn + mpn

M/M/1 System Balance Equations: Node 0) lp0 = mp1 Node 1) lp0 + mp2 = lp1 + mp1 Node 2) lp1 + mp3 = lp2 + mp2 : : : : Node n) lpn-1 + mpn+1 = lpn + mpn Determining state probabilities (pi): or,

M/M/1 System Additional Measures: Once all the probabilities (pi) are known, other performance measures can be calculated.

M/M/1 System Additional Measures: What is W? Wq?

M/M/s System – multiple servers 1 2 n-2 n-1 n … … m 2m sm sm sm Balance Equations: Node 0) lp0 = mp1 Node 1) lp0 + 2mp2 = lp1 + mp1 Node 2) lp1 + 3mp3 = lp2 + 2mp2 : : : : Node n) lpn-1 + smpn+1 = lpn + smpn

M/M/s System – multiple servers Determining state probabilities (pi): s p0 Lq 1 1-r 2 3 see Table 7.3 for other values of s.

Development of Balance Equations for Birth-Death Process M/M/1/k System – finite system capacity 1 2 k-2 k-1 k … Balance Equations:

Development of Balance Equations for Birth-Death Process M/M/1//n System – finite calling system population 1 2 N-2 N-1 N … Balance Equations:

M/G/1 Queue A system with a Poisson input, but some general distribution for service time. Only two characteristics of the service time need be known: the mean (1/m) and the variance (1/s2). P0 = 1 – r, where r = l/m Lq = L = r + Lq Wq = Lq / l W = Wq + 1 / m