### Abstract

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.

Original language | English |
---|---|

Title of host publication | Developments in Mathematics |

Publisher | Springer |

Pages | 327-342 |

Number of pages | 16 |

Volume | 58 |

ISBN (Electronic) | 9783030111021 |

ISBN (Print) | 9783030111014 |

DOIs | |

Publication status | Published - 3 Mar 2019 |

### Publication series

Name | Developments in Mathematics |
---|---|

Volume | 58 |

ISSN (Print) | 1389-2177 |

### Fingerprint

### Cite this

^{[}

^{m}

^{]}/1 queue. In

*Developments in Mathematics*(Vol. 58, pp. 327-342). (Developments in Mathematics; Vol. 58). Springer. https://doi.org/10.1007/978-3-030-11102-1_15

}

^{[}

^{m}

^{]}/1 queue. in

*Developments in Mathematics.*vol. 58, Developments in Mathematics, vol. 58, Springer, pp. 327-342. https://doi.org/10.1007/978-3-030-11102-1_15

**A combinatorial analysis of the M/M
^{[}
^{m}
^{]}
/1 queue.** / Mercankosk, Guven; Nair, Gopalan M.

Research output: Chapter in Book/Conference paper › Chapter

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

SN - 9783030111014

VL - 58

T3 - Developments in Mathematics

SP - 327

EP - 342

BT - Developments in Mathematics

PB - Springer

ER -

^{[}

^{m}

^{]}/1 queue. In Developments in Mathematics. Vol. 58. Springer. 2019. p. 327-342. (Developments in Mathematics). https://doi.org/10.1007/978-3-030-11102-1_15