Functional limit theorems for the number of busy servers in a G/G/∞ queue

2018 ◽  
Vol 55 (1) ◽  
pp. 15-29 ◽  
Author(s):  
Alexander Iksanov ◽  
Wissem Jedidi ◽  
Fethi Bouzeffour
Keyword(s):  
Weak Convergence ◽  
Integral Representation ◽  
Gaussian Process ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Open Problems ◽  
Skorokhod Space

Abstract We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J1-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.

Limit theorems for extremes with random sample size

1998 ◽  
Vol 30 (03) ◽  
pp. 777-806 ◽  
Author(s):  
Dmitrii S. Silvestrov ◽  
Jozef L. Teugels
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Sample Size ◽  
Extremal Processes ◽  
Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

Weak convergence in an appointment system

1975 ◽  
Vol 12 (1) ◽  
pp. 188-194 ◽  
Author(s):  
C. W. Anderson
Keyword(s):  
Weak Convergence ◽  
Convergence Theorem ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Service Facility ◽  
Appointment System ◽  
Weak Convergence Theorem ◽  
Carry Over

It is assumed that customers at a service facility have appointments at times 0,1,2, … for which they may be unpunctual by random amounts or may never arrive at all. A weak convergence theorem is proved for the process which counts the number of arrivals. This makes it possible to carry over the results of Iglehart and Whitt (1970a) to obtain heavy traffic functional limit theorems for queues with arrivals by appointment.

Multi-server assembly queues

1974 ◽  
Vol 11 (03) ◽  
pp. 629-632
Author(s):  
Michael A. Crane
Keyword(s):  
Weak Convergence ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Queue Size ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Functions ◽  
Wiener Processes ◽  
Multi Server

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

Limit theorems for extremes with random sample size

1998 ◽  
Vol 30 (3) ◽  
pp. 777-806 ◽  
Author(s):  
Dmitrii S. Silvestrov ◽  
Jozef L. Teugels
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Sample Size ◽  
Extremal Processes ◽  
Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

Measures in a functional spaces. Weak convergence, probability metrics. Functional limit theorems

2009 ◽  
pp. 241-270
Author(s):  
Dmytro Gusak ◽  
Alexander Kukush ◽  
Alexey Kulik ◽  
Yuliya Mishura ◽  
Andrey Pilipenko
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Probability Metrics

Multi-server assembly queues

1974 ◽  
Vol 11 (3) ◽  
pp. 629-632 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Weak Convergence ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Queue Size ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Functions ◽  
Wiener Processes ◽  
Multi Server

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

Weak convergence in an appointment system

1975 ◽  
Vol 12 (01) ◽  
pp. 188-194
Author(s):  
C. W. Anderson
Keyword(s):  
Weak Convergence ◽  
Convergence Theorem ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Service Facility ◽  
Appointment System ◽  
Weak Convergence Theorem ◽  
Carry Over

It is assumed that customers at a service facility have appointments at times 0,1,2, … for which they may be unpunctual by random amounts or may never arrive at all. A weak convergence theorem is proved for the process which counts the number of arrivals. This makes it possible to carry over the results of Iglehart and Whitt (1970a) to obtain heavy traffic functional limit theorems for queues with arrivals by appointment.

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.

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