1. 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

2. Rate (Transition) Diagram for Birth-Death Processl 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)

3. … … M/M/1 System l l l l l 1 2 n-2 n-1 n m m m m m1 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

4. 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,

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

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

7. M/M/s System – multiple servers1 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

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

9. 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:

10. 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:

11. 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