A Central Limit Theorem for Markov-Modulated Infinite-Server Queues

2013 ◽  
pp. 81-95 ◽  
Author(s):  
Joke Blom ◽  
Koen De Turck ◽  
Michel Mandjes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Infinite Server ◽  
Infinite Server Queues ◽  
Markov Modulated

scholarly journals ANALYSIS OF MARKOV-MODULATED INFINITE-SERVER QUEUES IN THE CENTRAL-LIMIT REGIME

2015 ◽  
Vol 29 (3) ◽  
pp. 433-459 ◽  
Author(s):  
Joke Blom ◽  
Koen De Turck ◽  
Michel Mandjes
Keyword(s):  
Central Limit ◽  
Background Process ◽  
Infinite Server ◽  
Deviation Matrix ◽  
Finite State ◽  
Transition Rate Matrix ◽  
Service Rates ◽  
Infinite Server Queues ◽  
Markov Modulated ◽  
Number Of Customers

This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix Q≡(qij)i,j=1d. Both arrival rates and service rates are depending on the state of the background process. The main contribution concerns the derivation of central limit theorems (CLTs) for the number of customers in the system at time t≥0, in the asymptotic regime in which the arrival rates λi are scaled by a factor N, and the transition rates qij by a factor Nα, with α∈ℝ+. The specific value of α has a crucial impact on the result: (i) for α>1 the system essentially behaves as an M/M/∞ queue, and in the CLT the centered process has to be normalized by √N; (ii) for α<1, the centered process has to be normalized by N1−α/2, with the deviation matrix appearing in the expression for the variance.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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