2. Multiple server queues In particular, we look at M/M/k Need to find steady state probabilities

3. M/M/k (k > 1) n =, for n = 0, 1, 2,.....  n = n , for n = 1, 2,..., k = k , for n = k, k+1,... kk 0 k+1 21 kk-1k-2   (k-1)  ... 3  Rate Diagram

4. M/M/k (cont.) State Rate In = Rate Out 0  P 1 = P 0 12  P 2 + P 0 = ( +  ) P 1 23  P 3 + P 1 = ( + 2  ) P 2....................... k-1 k  P k + P k-2 = { + (k-1)  } P k-1 k k  P k+1 + P k-1 = ( + k  ) P k k+1 k  P k+2 + P k = ( + k  ) P k+1.......................

5. M/M/k (cont.) Now, solve for P 1, P 2, P 3... in terms of P 0 P 1 = ( /  ) P 0 P 2 = ( / 2  ) P 1 = (1/2!)  ( /  ) 2 P 0 P 3 = ( / 3  ) P 2 = (1/3!)  ( /  ) 3 P 0......... P k = (1/k!)  ( /  ) k P 0 P k+1 = (1/k)  ( /  ) P k =

6. M/M/k (cont.) if 0  n  k if  k  n If < k 

7. M/M/k (cont.) Now solve for N q : Note,  = / k 

8. M/M/k (cont.) W = N q / (W: avg waiting time in Q) R = W + 1 /  (R: avg waiting time in sys.) N =  (W+ 1/  ) (N: avg # in the system) = N q + / 

9. Particular case : M/M/2  = / 2   P 0 = (1-  (1+   P n = 2  n (1-  (1+  n  1 W = N q / R = W + 1 /  N = N q + / 

10. Comparison of M/M/1 and M/M/2 2 counters. 2 types of jobs (internal and external). Exponential service time, avg 3 minutes. Internal: Poisson arrivals, 18 per hour External: Poisson arrivals, 15 per hour

11. Particular case : M/M/   ore servers than there are jobs ·Poisson distribution with parameter (  if 0  n  k

12. Performance of M/M/  For M/M/1: Same results also hold for M/G/ 