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if and only if L+ kc 1 and An kc 1 L+ n kc n kc
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For xed x and k, we now let n This gives
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+ where qi = limj P (Lj = i) It remains to nd the limiting probabilities qi These limiting probabilities can be obtained by a simple up- and downcrossing argument: the long-run fraction of customers nding k other customers in queue upon arrival equals the long-run fraction of customers leaving k other customers behind in queue when entering service This holds for any integer k 0 For k = 0 we also have that the long-run fraction of arrivals nding k other customers in queue equals the long-run fraction of arrivals who nd k + c other customers in the system This latter fraction equals the time-average probability pc+k by the PASTA property Hence we nd c
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Interchanging the order of summation in (9610), the result (969) now follows Asymptotic expansion It is also possible to give an asymptotic expansion for 1 Wq (x): 1 Wq (x) e ( 1)x where = as x , (9611)
( 1) c 1
with and as in (963) and (964) To prove this result, we x u with 0 u < D and let x run through (k 1)D + u for k = 1, 2, De ning
br (u) =
j =0
Qr j e (D u)
[ (D u)]j j!
for r = 0, 1, ,
we have by (969) that 1 Wq (x) = 1 bkc 1 (u) for x = (k 1)D + u
r Next consider the generating function B u (z) = r=0 (1 br (u))z Since the generating function of the convolution of two discrete sequences is the product of the generating functions of the separate sequences, it follows that
B u (z) = where Q(z) =
j j =0 Qj z
1 Q(z)e (D u)(z 1) , 1 z
c+j k=0 pk ,
Since Qj =
we nd after some algebra that e D(1 z)
c 1 k=0
Q(z) =
z c 1 z
pk (zk zc ) ,
P (z)
pk (zk zc ) =
(1 z)(1 zc e D(1 z) )
ALGORITHMIC ANALYSIS OF QUEUEING MODELS
where the latter equality uses (966) This leads to 1 zc e D(1 z) e u(1 z) B u (z) =
c 1 k=0
pk (zk zc ) /(1 z)
1 zc e D(1 z)
Next, by Theorem C1 in Appendix C and c e D(1 ) = 1, we nd 1 bj (u) e ( 1)u j 1 as j
Take now j = kc 1 and x = (k 1)D + u Then 1 bj (u) = 1 Wq (x) Since the equation c e D(1 ) = 1 implies (k 1)c = e ( 1)(k 1)D , we obtain 1 bkc 1 e ( 1)x ( 1) c 1 as k ,
which proves the desired result (9611) 962 The M/G/c Queue In this multi-server model with c servers the arrival process of customers is a Poisson process with rate and the service time S of a customer has a general probability distribution function B(t) It is assumed that the server utilization = E(S)/c is smaller than 1 The M/G/c queue with general service times permits no simple analytical solution, not even for the average waiting time Useful approximations can be obtained by the regenerative approach discussed in Section 921 In applying this approach to the multi-server queue, we encounter the dif culty that the number of customers left behind at a service completion epoch does not provide suf cient information to describe the future behaviour of the system In fact we need the additional information of the elapsed service times of the other services (if any) still in progress A full inclusion of this information in the state description would lead to an intractable analysis However, as an approximation, we will aggregate the information of the elapsed service times in such a way that the resulting approximate model enables us to carry through the regenerative analysis A closer look at the regenerative approach reveals that we need only a suitable approximation to the probability distribution of the time elapsed between service completions We now make the following approximation assumption with regard to the behaviour of the process at the service completion epochs Assumption 961 (approximation assumption) (a) If at a service completion epoch, k customers are left behind in the system with 1 k < c, then the time e e until the next service completion epoch is distributed as min(S1 , , Sk ), where