TY - CHAP

T1 - A combinatorial analysis of the M/M
[
m
]
/1 queue

AU - Mercankosk, Guven

AU - Nair, Gopalan M.

PY - 2019/3/3

Y1 - 2019/3/3

N2 -
Neuts’ Matrix Geometric Method makes use of the left-skip free characteristic of M/G/1-type Markov chains and determines the first passage distribution matrix G by solving a non-linear matrix equation. In this paper, we focus on the k-step first passage problem. In particular, we identify three associated matrices, namely the matrix G
k
, the conditional first passage probability matrix P
k
, and the first passage count matrix T
k
. The reformulation allows for combinatorial techniques. Specifically, we refer to an extension of Takács’ ballot theorem. We note that the matrix P
k
exhibits some ballot properties. In the case of the M/M
[
m
]
/1 queue, we establish the special structure of the count matrix T
k
using lattice path arguments. Furthermore, we obtain a closed-form expression for the G matrix, where the first passage probabilities are expressed in terms of generalized hypergeometric functions.

AB -
Neuts’ Matrix Geometric Method makes use of the left-skip free characteristic of M/G/1-type Markov chains and determines the first passage distribution matrix G by solving a non-linear matrix equation. In this paper, we focus on the k-step first passage problem. In particular, we identify three associated matrices, namely the matrix G
k
, the conditional first passage probability matrix P
k
, and the first passage count matrix T
k
. The reformulation allows for combinatorial techniques. Specifically, we refer to an extension of Takács’ ballot theorem. We note that the matrix P
k
exhibits some ballot properties. In the case of the M/M
[
m
]
/1 queue, we establish the special structure of the count matrix T
k
using lattice path arguments. Furthermore, we obtain a closed-form expression for the G matrix, where the first passage probabilities are expressed in terms of generalized hypergeometric functions.

KW - Ballot theorems

KW - Combinatorial techniques

KW - Generalized hypergeometric functions

KW - Lattice paths

KW - M/M /1 queue

UR - http://www.scopus.com/inward/record.url?scp=85062903425&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-11102-1_15

DO - 10.1007/978-3-030-11102-1_15

M3 - Chapter

AN - SCOPUS:85062903425

SN - 9783030111014

VL - 58

T3 - Developments in Mathematics

SP - 327

EP - 342

BT - Developments in Mathematics

PB - Springer

ER -