scholarly journals Necessary conditions in limit theorems for cumulative processes

2002 ◽  
Vol 98 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Peter W. Glynn ◽  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Necessary Conditions ◽  
Cumulative Processes

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

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.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (03) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

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.

A Central Limit Theorem for Cumulative Processes

1994 ◽  
Vol 26 (01) ◽  
pp. 104-121 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Independent Random Variables ◽  
Cumulative Processes

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.

scholarly journals Limit theorems for cumulative processes

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

Limit theorems for generalized single server queues

1974 ◽  
Vol 6 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Queueing System ◽  
Single Server ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Convergence Results ◽  
Cumulative Processes ◽  
Number Of Customers ◽  
Functional Central Limit

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.

Some Generalized Central Limit Theorems for Weighted Sums with Infinite Variance

1989 ◽  
Vol 38 (1-2) ◽  
pp. 27-42 ◽  
Author(s):  
André Adler ◽  
Andrew Rosalsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Necessary Conditions ◽  
Random Variables ◽  
Weighted Sums ◽  
Infinite Variance ◽  
Famous Result ◽  
Special Case

For i.i.d. random variables { Y, Yn, n⩾ 1} with EY2=∞ and nonzero constants { an, n⩾1}, sufficient and, separately, necessary conditions are given for { an Yn, n⩾1} to obey a generalized central limit theorem [Formula: see text] for suitable constants { An, n⩾1} and { Bn > 0, n⩾1}. The norming constants { Bn, n⩾1} are defined using the sequence [Formula: see text] and the distribution of Y. Moreover, it is shown that if p{| Y|> y} is regularly varying with exponent-2, then the centering constants may be taken to be [Formula: see text]. A famous result of Feller (1935), Khintchine (1935), and Lévy (1935) is obtained in the special case an ≡ 1.

Limit theorems for generalized single server queues

1974 ◽  
Vol 6 (01) ◽  
pp. 159-174 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Queueing System ◽  
Single Server ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Convergence Results ◽  
Cumulative Processes ◽  
Number Of Customers ◽  
Functional Central Limit

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.

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