Analysis of state-dependent discrete-time queue with system disaster

2019 ◽  
Vol 53 (5) ◽  
pp. 1915-1927
Author(s):  
Ramupillai Sudhesh ◽  
Arumugam Vaithiyanathan
Keyword(s):  
Discrete Time ◽  
Busy Period ◽  
System Size ◽  
Time Dependent ◽  
System Effect ◽  
Period Distribution ◽  
State Dependent ◽  
Discrete Time Queue ◽  
Formal Power ◽  
Time Dependent System

An explicit expression for time-dependent system size probabilities is obtained for the general state-dependent discrete-time queue with system disaster. Using generating function for the nth state transient probabilities, the underlying difference equation of system size probabilities are transformed into three-term recurrence relation which is then expressed as a continued fraction. The continued fractions are converted into formal power series which yield the time-dependent system size probabilities in closed form. Further, the busy period distribution is obtained for the considered model. As a special case, the system size probabilities and busy period distribution of Geo/Geo/1 queue are deduced. Finally, numerical illustrations are presented to visualize the system effect for various values of the parameters.

On the expected number of crossings of a level in certain stochastic processes

1970 ◽  
Vol 7 (3) ◽  
pp. 766-770 ◽  
Author(s):  
P. B. M. Roes
Keyword(s):  
Present Method ◽  
Busy Period ◽  
Time Dependent ◽  
Renewal Processes ◽  
Time Process ◽  
Expected Number ◽  
Original Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].

Busy period distribution in state-dependent queues

1987 ◽  
Vol 2 (3) ◽  
pp. 285-305 ◽  
Author(s):  
C. Knessl ◽  
B. J. Matkowsky ◽  
Z. Schuss ◽  
C. Tier
Keyword(s):  
Busy Period ◽  
Period Distribution ◽  
State Dependent

On the expected number of crossings of a level in certain stochastic processes

1970 ◽  
Vol 7 (03) ◽  
pp. 766-770 ◽  
Author(s):  
P. B. M. Roes
Keyword(s):  
Present Method ◽  
Busy Period ◽  
Time Dependent ◽  
Renewal Processes ◽  
Time Process ◽  
Expected Number ◽  
Original Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as opposed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3].

scholarly journals A study on some infinite server queues in discrete-time

10.26524/cm78 ◽  
2020 ◽  
Vol 4 (2) ◽  
Author(s):  
Syed Tahir Hussainy ◽  
Lokesh D
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Shot Noise ◽  
Busy Period ◽  
Transient State ◽  
System Size ◽  
Differentiation Formula ◽  
Work Analysis ◽  
Shot Noise Process ◽  
Infinite Server Queues

This work analysis some discrete-time queueing mechanisms with infinitely many servers.By using a shot noise process, general results on the system size in discrete-time are given both in transient state and in steady state. For this we use the classical differentiation formula of F´a di Bruno. First two moments of the system size and distribution of the busy period of the system are also computed.

scholarly journals Elliptic Solutions of Dynamical Lucas Sequences

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 183
Author(s):  
Michael J. Schlosser ◽  
Meesue Yoo
Keyword(s):  
Discrete Time ◽  
Time Dependent ◽  
Fibonacci Polynomials ◽  
Lucas Sequences ◽  
Elliptic Solutions

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler–Cassini identity.

Sign in / Sign up

Export Citation Format

Share Document