ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

2012 ◽  
Vol 26 (2) ◽  
pp. 221-244 ◽  
Author(s):  
M. L. Chaudhry ◽  
S. K. Samanta ◽  
A. Pacheco
Keyword(s):  
Length Distribution ◽  
Renewal Theory ◽  
Computationally Efficient ◽  
Form Analysis ◽  
Matrix Geometric Method ◽  
Alternative Approach ◽  
The Matrix ◽  
Sojourn Time Distribution ◽  
System Length ◽  
Markov Renewal Theory

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (03) ◽  
pp. 614-620
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn),n≧ 0} be the standardJ–Xprocess of Markov renewal theory. Suppose {Jn,n≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that ifconverges in distribution, wherean, bn>0 (bn→∞) are real constants, then the limit lawFmust be stable. SupposeQ(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn),n≧ 0}. Then then-fold convolution,Q∗n(bnx + anbn), converges in distribution toF(x)Π if and only ifconverges in distribution toF. Π is the matrix of stationary transition probabilities of {Jn,n≧ 0}. Sufficient conditions on theHi's are given for the convergence of the sequence of semi-Markov matrices toF(x)Π, whereFis stable.

scholarly journals The MX/G/1 queue with queue length dependent service times

2001 ◽  
Vol 14 (4) ◽  
pp. 399-419 ◽  
Author(s):  
Bong Dae Choi ◽  
Yeong Cheol Kim ◽  
Yang Woo Shin ◽  
Charles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Time Distribution ◽  
Length Distribution ◽  
Renewal Theory ◽  
Waiting Time Distribution ◽  
Queue Length Distribution ◽  
Virtual Waiting Time ◽  
Service Times ◽  
Mean Queue Length ◽  
Markov Renewal Theory

We deal with the MX/G/1 queue where service times depend on the queue length at the service initiation. By using Markov renewal theory, we derive the queue length distribution at departure epochs. We also obtain the transient queue length distribution at time t and its limiting distribution and the virtual waiting time distribution. The numerical results for transient mean queue length and queue length distributions are given.

scholarly journals Quasistochastic matrices and Markov renewal theory

2014 ◽  
Vol 51 (A) ◽  
pp. 359-376
Author(s):  
Gerold Alsmeyer
Keyword(s):  
Probabilistic Approach ◽  
Nonnegative Matrix ◽  
Renewal Theory ◽  
Distribution Functions ◽  
Markov Renewal Equation ◽  
Markov Random Walk ◽  
The Matrix ◽  
Left And Right ◽  
Irreducible Nonnegative Matrix ◽  
Markov Renewal Theory

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

scholarly journals Quasistochastic matrices and Markov renewal theory

2014 ◽  
Vol 51 (A) ◽  
pp. 359-376 ◽  
Author(s):  
Gerold Alsmeyer
Keyword(s):  
Probabilistic Approach ◽  
Nonnegative Matrix ◽  
Renewal Theory ◽  
Distribution Functions ◽  
Markov Renewal Equation ◽  
Markov Random Walk ◽  
The Matrix ◽  
Left And Right ◽  
Irreducible Nonnegative Matrix ◽  
Markov Renewal Theory

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Limit theorems for sums of a sequence of random variables defined on a Markov chain

1977 ◽  
Vol 14 (3) ◽  
pp. 614-620 ◽  
Author(s):  
David B. Wolfson
Keyword(s):  
Limit Theorems ◽  
Transition Probabilities ◽  
Sufficient Conditions ◽  
Renewal Theory ◽  
Strong Mixing ◽  
Mixing Condition ◽  
The Matrix ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Markov Renewal Theory

Let {(Jn, Xn), n ≧ 0} be the standard J–X process of Markov renewal theory. Suppose {Jn, n ≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that if converges in distribution, where an, bn > 0 (bn → ∞) are real constants, then the limit law F must be stable. Suppose Q(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn), n ≧ 0}. Then the n-fold convolution, Q∗n(bnx + anbn), converges in distribution to F(x)Π if and only if converges in distribution to F. Π is the matrix of stationary transition probabilities of {Jn, n ≧ 0}. Sufficient conditions on the Hi's are given for the convergence of the sequence of semi-Markov matrices to F(x)Π, where F is stable.

scholarly journals Theory of semiregenerative phenomena

1988 ◽  
Vol 25 (A) ◽  
pp. 257-274
Author(s):  
N. U. Prabhu
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Renewal Process ◽  
Renewal Theory ◽  
Markov Renewal Process ◽  
Discrete Time Case ◽  
Markov Renewal Theory

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

scholarly journals An Alternative Approach for the Detection of Ethephon (2-Chlorethylphosphonic acid) Residues in Apples

2006 ◽  
Vol 1 (4) ◽  
pp. 1934578X0600100
Author(s):  
Dietmar Kröpfl ◽  
Klaus Schweiger ◽  
Franz Siegfried Wagner ◽  
Elke Prettner
Keyword(s):  
Internal Standard ◽  
Good Method ◽  
Low Ph ◽  
Measurement Precision ◽  
Direct Extraction ◽  
Selected Ion Monitoring ◽  
Method Performance ◽  
Alternative Approach ◽  
The Matrix ◽  
One Step

A rapid and simple GC/MS method to detect ethephon residues in apples is introduced using an internal standard and direct extraction of the analytes from the matrix at low pH with ethyl acetate. A one step derivatization procedure has been established using N-tert-butyldimethylsilyl-N-methyltrifluoracetamide (MTBSTFA) prior to GC/MS analysis performed in selected ion monitoring mode (SIM). Optimization of the derivatization and GC- parameters led to good method performance and measurement precision. Recovery experiments showed recovery rates ranging from 80.8 to 95.5% in the concentration range were investigated. The presented method provides significant improvements in laboratory safety, sample throughput and instrumentation cost compared with other methods to detect ethephon residues in apples.

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