Classification of Markov Processes of Matrix M/G/1 type with a Tree Structure and its Applications to the MMAP[K]/G[K]/1 Queues
نویسنده
چکیده
This paper studies the classification problem of discrete time and continuous time Markov processes of matrix M/G/1 type with a tree structure. It is shown that the Perron-Frobenius eigenvalue of a nonnegative matrix provides information for a complete classification of the Markov process of interest. A computational method is developed to find whether a Markov process of matrix M/G/1 type with a tree structure is positive recurrent, null recurrent, or transient. The method is then used to study the impact of the last-come-first-served general preemptive resume (LCFSGPR) service discipline on the stability of a MAP/PH/1 queue. Two sufficient conditions are identified for the positive recurrence and transience of the Markov processes of interest, respectively. As an example, the results are used to show that the discrete time (and continuous time) MMAP[K]/G[K]/1 queue with a work conserving service discipline is stable if and only if its traffic intensity is less than one, unstable if its traffic intensity is larger than one.
منابع مشابه
Classification of Markov Processes of Matrix M/G/1 type with a Tree Structure and its Applications to the MMAP[K]/G[K]/1 Queue
The purpose of this paper is to study the classification problem of discrete time and continuous time Markov processes of matrix M/G/1 type with a tree structure. We begin this paper by developing a computational method to find whether a Markov process of matrix M/G/1 type with a tree structure is positive recurrent, null recurrent, or transient. The method is then used to study the impact of t...
متن کاملRelationship between Coefficients of Characteristic Polynomial and Matching Polynomial of Regular Graphs and its Applications
ABSTRACT. Suppose G is a graph, A(G) its adjacency matrix and f(G, x)=x^n+a_(n-1)x^(n-1)+... is the characteristic polynomial of G. The matching polynomial of G is defined as M(G, x) = x^n-m(G,1)x^(n-2) + ... where m(G,k) is the number of k-matchings in G. In this paper, we determine the relationship between 2k-th coefficient of characteristic polynomial, a_(2k), and k-th coefficient of matchin...
متن کاملAsymptotic behaviour of associated primes of monomial ideals with combinatorial applications
Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. We say that $I$ satisfies the persistence property if $mathrm{Ass}_R(R/I^k)subseteq mathrm{Ass}_R(R/I^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{Ass}_R(R/I)$ denotes the set of associated prime ideals of $I$. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring $R=K[x_1,ld...
متن کاملSIGNLESS LAPLACIAN SPECTRAL MOMENTS OF GRAPHS AND ORDERING SOME GRAPHS WITH RESPECT TO THEM
Let $G = (V, E)$ be a simple graph. Denote by $D(G)$ the diagonal matrix $diag(d_1,cdots,d_n)$, where $d_i$ is the degree of vertex $i$ and $A(G)$ the adjacency matrix of $G$. The signless Laplacianmatrix of $G$ is $Q(G) = D(G) + A(G)$ and the $k-$th signless Laplacian spectral moment of graph $G$ is defined as $T_k(G)=sum_{i=1}^{n}q_i^{k}$, $kgeqslant 0$, where $q_1$,$q_2$, $cdots$, $q_n$ ...
متن کاملLongest Path in Networks of Queues in the Steady-State
Due to the importance of longest path analysis in networks of queues, we develop an analytical method for computing the steady-state distribution function of longest path in acyclic networks of queues. We assume the network consists of a number of queuing systems and each one has either one or infinite servers. The distribution function of service time is assumed to be exponential or Erlang. Fu...
متن کامل