Some limit theorems for cumulative processes with applications to sojourn times

1987 ◽  
Vol 6 (2) ◽  
pp. 83-89
Author(s):  
V.G. Kulkarni
1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.


1972 ◽  
Vol 9 (03) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt

The stable GI/G/s queue (ρ &lt; 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ &lt; 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.


1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.


1993 ◽  
Vol 47 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt

1974 ◽  
Vol 6 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Austin J. Lemoine

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.


1974 ◽  
Vol 6 (01) ◽  
pp. 159-174 ◽  
Author(s):  
Austin J. Lemoine

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.


1991 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt

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