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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
Multiple server queues

  • In particular, we look at M/M/k

  • Need to find steady state probabilities


M m k k 1

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)


M m k cont
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

.... ...................


M m k cont1
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 =


M m k cont2
M/M/k (cont.)

If l< km =>

if 0 £ n £ k

if k £ n


M m k cont3
M/M/k (cont.)

Now solve for Nq: Note, r = l/ km


M m k cont4
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


Particular case m m 2
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 =


Comparison of m m 1 and m m 2
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


Particular case m m
Particular case : M/M/

if 0 £ n £ k

  • More servers than there are jobs

  • Poisson distribution with parameter (l/m)


Performance of m m a
Performance of M/M/a

  • For M/M/1:

  • Same results also hold for M/G/a


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