MULTI-SERVER QUEUES WITH POISSON INPUT

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e e S1 , , Sk are independent random variables that have the equilibrium excess distribution function

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Be (t) =

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1 E(S)

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{1 B(x)}dx,

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t 0,

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as probability distribution function (b) If at a service completion epoch, k customers are left behind in the system with k c, then the time until the next service completion is distributed as S/c, where S denotes the original service time of a customer This approximation assumption can be motivated as follows First, if not all c servers are busy, the M/G/c queueing system may be treated as an M/G/ queueing system in which a free server is immediately provided to each arriving customer For the M/G/ queue in statistical equilibrium it was shown by Tak cs a (1962) that the remaining service time of any busy server is distributed as the residual life in a renewal process with the service times as the interoccurrence times The same is true for the M/G/1 queue; see formula (9232) The equilibrium excess distribution of the service time is given by Be (t); see Theorem 825 Second, if all of the c servers are busy, then the M/G/c queue may be approximated by an M/G/1 queue in which the single server works c times as fast as each of the c servers in the original multi-server system It is pointed out that the approximation assumption holds exactly for both the case of the c = 1 server and the case of exponentially distributed service times Approximations to the state probabilities Under the approximation assumption the recursion scheme derived in Section 921 for the M/G/1 queue can be extended to the M/G/c queue to yield approximations app pj to the state probabilities pj These approximations are given in the next theorem, whose lengthy proof may be skipped at rst reading The approximation to the state probabilities implies an approximation to the waiting-time probabilities The latter approximation is discussed in Exercise 911 Theorem 961 Under the approximation assumption, pj pj

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(c )j app p , j! 0

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j = 0, 1, , c 1,

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j k=c

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(9612)

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= aj c pc 1 +

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bj k pk ,

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j = c, c + 1, ,

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(9613)

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where the constants an and bn are given by an =

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{1 Be (t)}c 1 {1 B(t)}e t

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( t)n dt, n!

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n = 0, 1, ,

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386 bn = Proof

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ALGORITHMIC ANALYSIS OF QUEUEING MODELS

{1 B(ct)}e t

( t)n dt, n!

n = 0, 1,

In the same way as in the proof of Theorem 921, we nd

app pj

app p0 A0j

pk Akj ,

j = 1, 2, ,

(9614)

where the constant Akj is de ned by Akj = the expected amount of time that j customers are present during the time until the next service completion epoch when a service has just been completed with k customers left behind in the system By the same argument as used to derive (927), we nd under the approximation assumption that Akj =

{1 B(ct)}e t

( t)j k dt, (j k)!

k c and j k

(9615)

However, the problem is to nd a tractable expression for Akj when 0 k c 1 An explicit expression for Akj involves a multidimensional integral when 0 k c 1 Fortunately, this dif culty can be circumvented by de ning, for any 1 k c and j k, the probability Mkj (t) by Mkj (t) = P {j k customers arrive during the next t time units and the service of none of these customers is completed in the next t time units when only c k servers are available for the new arrivals} Then, using the approximation assumption, Akj =

{1 Be (t)}k Mkj (t) dt,

1 k c 1, j k

(9616)

Further, we have A0j =

{1 B(t)}M1j (t) dt,

j 1

The de nition of Mkj (t) implies that Mkk (t) = e t , k 1 and Mcj (t) = e t ( t)j c , (j c)! j c

Next we derive a differential equation for Mkj (t) when j > k By conditioning on what may happen in the rst t time units, we nd for any 1 k c 1 and j > k that Mkj (t + t) = (1 t)Mkj (t) + t{1 B(t)}Mk+1,j (t) + o( t), t > 0