By Gordon F. Newell (auth.)
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Lb) is approximately correct, but to obtain a second approximation, one must proceed differently from the above. 2) then there is a scaling of coordinates L and T , with T of order O. 52 such that, for sufficiently large t, gt approaches some limiting form. 3b). 5) has been arranged so that both sides are derivatives with respect to £1. 2). 2). 8) + ~ 2110Cllil - flO + fll + exp(Clli l ) - 1 - Cl l \ exp(Cllc l ) - 1 [eXP(_Clli l ) Clg(xO'O ;t) ~--~----- Cl l axe This still does not describe an explicit solution for it only relates g(xO,i;t) to an as yet unknown function the probability density at zero queue.
Lj* The new terms in should automatically force the system to do what the boundary conditions of section 4 were supposed to guarantee. 1). Each term on the right-hand side integrates out to a surface term where f vanishes. 5) near the boundary n. 7) f this large term must be cancelled by a large term from the second derivatives. The solution positive. 8). 8) for ~ = ~-l. 5) in the vicinity of the edges is much more difficult, even the asymptotic solution for nk ~ o. 5) if we ~ ~ 0, will show various non-analytic behavior near the edges postulate that the total probability near edges is negligible compared with that near the boundary surfaces.
T) J n. J Thus, instead of the condition would now satisfy A negative queue means that D. (t) - n. (t) > 0 J J servers are idle. We will also allow pens, we interpret Qj+l(t) Qj+l(t) - c j + 1 to exceed but, when this hap- as the number of customers in the jth service who have completed service but had no place to go. They remain 33 in the jth service but block the server from serving any new customers. Note that we still retain the symmetry of section 3 because a server is idle either if it has no customer or has a blocking customer.
Approximate Behavior of Tandem Queues by Gordon F. Newell (auth.)