On Lattice Path Counting and the Random Product Representation, with Applications to the Er/M/1 Queue and the M/Er/1 Queue

2018 ◽  
Vol 21 (4) ◽  
pp. 1119-1149 ◽  
Author(s):  
Xiaoyuan Liu ◽  
Brian Fralix
Keyword(s):  
Lattice Path ◽  
Product Representation ◽  
Random Product ◽  
Path Counting ◽  
Lattice Path Counting

scholarly journals A comparative analysis of the successive lumping and the lattice path counting algorithms

2016 ◽  
Vol 53 (1) ◽  
pp. 106-120 ◽  
Author(s):  
Michael N. Katehakis ◽  
Laurens C. Smit ◽  
Floske M. Spieksma
Keyword(s):  
Lattice Path ◽  
Queueing Models ◽  
Rate Matrix ◽  
Steady State Distribution ◽  
State Spaces ◽  
Finite State ◽  
Counting Algorithms ◽  
Path Counting ◽  
Lattice Path Counting ◽  
Numerical Complexity

Abstract This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al. (2009) and Van Leeuwaarden and Winands (2006). The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models considered in Van Leeuwaarden et al. (2009). The main findings are threefold. First, when both methods are applicable, the SL-based algorithms outperform the lattice path counting algorithm (LPCA). Second, there are important classes of problems (for example, models with (level) nonhomogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. Third, another main advantage of SL algorithms over lattice path counting is that the former includes a method to compute the steady state distribution using this rate matrix.

Lattice Path Counting and Applications (Sri Gopal Mohanty)

SIAM Review ◽  
1983 ◽  
Vol 25 (4) ◽  
pp. 592-593
Author(s):  
L. F. Takacs
Keyword(s):  
Lattice Path ◽  
Path Counting ◽  
Lattice Path Counting

scholarly journals On Lattice Path Counting by Major Index and Descents

1993 ◽  
Vol 14 (1) ◽  
pp. 43-51 ◽  
Author(s):  
C. Krattenthaler ◽  
S.G. Mohanty
Keyword(s):  
Lattice Path ◽  
Major Index ◽  
Path Counting ◽  
Lattice Path Counting

scholarly journals Talmudic lattice path counting

1994 ◽  
Vol 68 (1) ◽  
pp. 215-217
Author(s):  
Jane Friedman ◽  
Ira Gessel ◽  
Doron Zeilberger
Keyword(s):  
Lattice Path ◽  
Path Counting ◽  
Lattice Path Counting

Lattice Path Counting and Applications.

1980 ◽  
Vol 143 (4) ◽  
pp. 524
Author(s):  
M. Knott ◽  
Sri Gopal Mohanty
Keyword(s):  
Lattice Path ◽  
Path Counting ◽  
Lattice Path Counting

scholarly journals Quasi-Birth-and-Death Processes, Lattice Path Counting, and Hypergeometric Functions

2009 ◽  
Vol 46 (02) ◽  
pp. 507-520 ◽  
Author(s):  
Johan S. H. van Leeuwaarden ◽  
Mark S. Squillante ◽  
Erik M. M. Winands
Keyword(s):  
Hypergeometric Functions ◽  
Lattice Path ◽  
Explicit Solutions ◽  
Rate Matrix ◽  
Birth And Death Processes ◽  
Counting Problem ◽  
Classical Models ◽  
Associated Matrix ◽  
Path Counting ◽  
Lattice Path Counting

In this paper we consider a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G . The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.

scholarly journals Quasi-Birth-and-Death Processes, Lattice Path Counting, and Hypergeometric Functions

2009 ◽  
Vol 46 (2) ◽  
pp. 507-520 ◽  
Author(s):  
Johan S. H. van Leeuwaarden ◽  
Mark S. Squillante ◽  
Erik M. M. Winands
Keyword(s):  
Hypergeometric Functions ◽  
Lattice Path ◽  
Explicit Solutions ◽  
Rate Matrix ◽  
Birth And Death Processes ◽  
Counting Problem ◽  
Classical Models ◽  
Associated Matrix ◽  
Path Counting ◽  
Lattice Path Counting

In this paper we consider a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.

Sign in / Sign up

Export Citation Format

Share Document