Light Traffic Limits of Sojourn Time Distributions in Markovian Queueing Networks

Light-Traffic Analysis for Queues with Spatially Distributed Arrivals

“…This approach seems to be very promising for deriving (higher-order) light-traffic approximations for queues with a (non-Poissonian) arrival process possessing some special dependence structure which still allows us to calculate limits of higher-order Palm distributions and factorial moment measures when the arrival intensity a tends to zero. In Baccelli and Bremaud (1993), conditions different from those in Blaszczyszyn (1995) and Reiman and Simon (1989) have been given for the validity of analogous formulas for light-traffic derivatives, where in Baccelli and Bremaud (1993) mostly the case has been considered that the input is a general stationary marked point process. The crucial step of the approach given in Baccelli and Bremaud (1993) is to show that a certain functional g is nonnegative almost surely (see condition (i) of Theorem 1' in Baccelli and Bremaud (1993)) and, then, to use Campbell's formula for stationary marked point processes.…”

“…Observe that we are in the same framework as Reiman and Simon (1989), that is, we are dealing with a marked Poisson process {(! ", ZJ) on U, with markings Z,, = iX,,,SJ, where X^ denotes the position (relative to the position of the server at time T^) and S^ the service time of the customer arriving at r,,.…”

“…This will be done in a separate paper. Also if heavy traffic results would be available, these Taylor-expansions could be used to get via an appropriate interpolation, approximation formulas for an arbitrary positive arrival rate a; see Fendick and Whitt (1989), Reiman and Simon (1988a), Whitt (1989). The behavior of the greedy server in heavy traffic is a very interesting (open) problem.…”

“…We will show by using the following Theorem 2 (which is an extension of Corollary 5.2 in Blaszczyszyn (1995)) that the first and second (right-hand) derivatives f^HO) and f^^KO) of / at a = 0 exist, under the condition that (2.3) holds and that the fifth moment e^ of service times is finite. Moreover, Corollary 5.2 in Blaszczyszyn (1995) provides a way to calculate these derivatives via the functions h and g, defined above (see also Theorem 2 of Reiman and Simon (1989), where an additional "admissibility" condition is assumed). Related results for the "usual" multi-server queue (without walking times) have recently been derived in Blaszczyszyn, Frey and Schmidt (1995); see also the survey given in Blaszczyszyn, Rolski and Schmidt (1995).…”

“…In our search for light-traffic results, we are led by Baccelli and Bremaud (1993), Blaszczyszyn (1995) and Reiman and Simon (1989) (see also Baccelli and Bremaud (1994), Reiman and Simon (1988b), Sigman (1992), Simon (1993), Whitt (1988Whitt ( , 1989). In particular, we will use an extension of Corollary 5.2 in Blaszczyszyn (1995) to the case of an independently marked Poisson process.…”