Embedded renewal processes in the GI/G/s queue

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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Embedded renewal processes in the GI/G/s queue

Journal of Applied Probability ◽

10.2307/3212333 ◽

1972 ◽

Vol 9 (3) ◽

pp. 650-658 ◽

Cited By ~ 35

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

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Functional limit theorems for the queue GI/G/1 in light traffic

Advances in Applied Probability ◽

10.1017/s0001867800037940 ◽

1971 ◽

Vol 3 (02) ◽

pp. 269-281 ◽

Cited By ~ 11

Author(s):

Donald L. Iglehart

Keyword(s):

Service Time ◽

Limit Theorems ◽

Queueing System ◽

Deterministic System ◽

Traffic Intensity ◽

Random Vectors ◽

Functional Limit ◽

Light Traffic ◽

Functional Limit Theorems ◽

Natural Measure

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t 0 = 0, finds a free server, and experiences a service time v 0. The nth customer arrives at time t n and experiences a service time v n . Let the interarrival times t n - t n-1 = u n , n ≧ 1, and define the random vectors X n = (v n-1, u n ), n ≧ 1. We assume the sequence of random vectors {X n : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{u n } = λ-1 and E{v n } = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v 0 2} < ∞ and that the deterministic system in which both v n and u n are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.

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Functional limit theorems for the queue GI/G/1 in light traffic

Advances in Applied Probability ◽

10.2307/1426171 ◽

1971 ◽

Vol 3 (2) ◽

pp. 269-281 ◽

Cited By ~ 24

Author(s):

Donald L. Iglehart

Keyword(s):

Service Time ◽

Limit Theorems ◽

Queueing System ◽

Deterministic System ◽

Traffic Intensity ◽

Random Vectors ◽

Functional Limit ◽

Light Traffic ◽

Functional Limit Theorems ◽

Natural Measure

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t0 = 0, finds a free server, and experiences a service time v0. The nth customer arrives at time tn and experiences a service time vn. Let the interarrival times tn - tn-1 = un, n ≧ 1, and define the random vectors Xn = (vn-1, un), n ≧ 1. We assume the sequence of random vectors {Xn : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{un} = λ-1 and E{vn} = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v02} < ∞ and that the deterministic system in which both vn and un are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.

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