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Multiple server queues

Multiple server queues. In particular, we look at M/M/k Need to find steady state probabilities. l. l. , for n = 0, 1, 2,. =. n. m. = n. , for n = 1, 2,..., k. m. n. = k. , for n = k, k+1,. m. Rate Diagram. l. l. l. l. l. l. 0. 1. 2. 3. k-2. k-1.

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Multiple server queues

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

  2. l l , for n = 0, 1, 2,..... = n m = n , for n = 1, 2,..., k m n = k , for n = k, k+1,... m Rate Diagram l l l l l l ... ... 0 1 2 3 k-2 k-1 k k+1 m 2m 3m m m m (k-1) k k M/M/k (k > 1)

  3. M/M/k (cont.) State Rate In = Rate Out 0 mP1 = lP0 1 2mP2 + lP0 = (l + m) P1 2 3mP3 + lP1 = (l + 2m) P2 .... ................... k-1 kmPk + lPk-2 = {l + (k-1)m} Pk-1 k kmPk+1 + lPk-1 = (l + km) Pk k+1 kmPk+2 + lPk = (l + km) Pk+1 .... ...................

  4. M/M/k (cont.) Now, solve for P1 , P2, P3... in terms of P0 P1 = (l/ m) P0 P2 = (l/ 2m) P1 = (1/2!) × (l/ m)2 P0 P3 = (l/ 3m) P2 = (1/3!) × (l/ m)3 P0 ......... Pk = (1/k!) × (l/ m)k P0 Pk+1 = (1/k) × (l/ m) Pk =

  5. M/M/k (cont.) If l< km => if 0 £ n £ k if k £ n

  6. M/M/k (cont.) Now solve for Nq: Note, r = l/ km

  7. M/M/k (cont.) W= Nq / l (W: avg waiting time in Q) R = W+ 1 / m (R: avg waiting time in sys.) N = l (W+ 1/m) (N: avg # in the system) = Nq + l/ m

  8. Particular case : M/M/2 • r = l/ 2m • P0 = (1- r)/ (1+ r) • Pn = 2 rn(1-r)/ (1+ r), n  1 W= Nq / l = R = W+ 1 / m N = Nq + l/ m =

  9. 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

  10. Particular case : M/M/ if 0 £ n £ k • More servers than there are jobs • Poisson distribution with parameter (l/m)

  11. Performance of M/M/a • For M/M/1: • Same results also hold for M/G/a

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