Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

1990 ◽  
Vol 27 (3) ◽  
pp. 684-692 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Length Distribution ◽  
Direct Proof ◽  
Interarrival Time ◽  
Convex Order ◽  
Queue Length Distribution ◽  
Virtual Waiting Time ◽  
Loss Probabilities ◽  
Waiting Time Process

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

1990 ◽  
Vol 27 (03) ◽  
pp. 684-692
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Length Distribution ◽  
Direct Proof ◽  
Interarrival Time ◽  
Convex Order ◽  
Queue Length Distribution ◽  
Virtual Waiting Time ◽  
Loss Probabilities ◽  
Waiting Time Process

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ &lt; 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

scholarly journals New Approach for Finding Basic Performance Measures of Single Server Queue

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Siew Khew Koh ◽  
Ah Hin Pooi ◽  
Yi Fei Tan
Keyword(s):  
Stationary State ◽  
Queue Length ◽  
Length Distribution ◽  
Cumulative Distribution ◽  
System Capacity ◽  
Interarrival Time ◽  
Single Server ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue

Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density functionf(t)and the cumulative distribution functionF(t)of the interarrival time are such that the ratef(t)/1-F(t)tends to a constant ast→∞, and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state.

Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue

1981 ◽  
Vol 13 (03) ◽  
pp. 619-630 ◽  
Author(s):  
Yukio Takahashi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Length Distribution ◽  
Waiting Time Distribution ◽  
Queue Length Distribution ◽  
Multiserver Queues ◽  
The Matrix ◽  
Rates Of Decay

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distributionW(x) has an asymptotically exponential tail, i.e., 1 –W(x) ∽Ke–ckx. The parameter k is the unique positive number satisfyingT*(ck)S*(–k) = 1, whereT*(s) andS*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η =T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constantKis not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.

scholarly journals The MX/G/1 queue with queue length dependent service times

2001 ◽  
Vol 14 (4) ◽  
pp. 399-419 ◽  
Author(s):  
Bong Dae Choi ◽  
Yeong Cheol Kim ◽  
Yang Woo Shin ◽  
Charles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Time Distribution ◽  
Length Distribution ◽  
Renewal Theory ◽  
Waiting Time Distribution ◽  
Queue Length Distribution ◽  
Virtual Waiting Time ◽  
Service Times ◽  
Mean Queue Length ◽  
Markov Renewal Theory

We deal with the MX/G/1 queue where service times depend on the queue length at the service initiation. By using Markov renewal theory, we derive the queue length distribution at departure epochs. We also obtain the transient queue length distribution at time t and its limiting distribution and the virtual waiting time distribution. The numerical results for transient mean queue length and queue length distributions are given.

scholarly journals Repairable Queue with Non-exponential Interarrival Time and Variable Breakdown Rates

2018 ◽  
Vol 7 (2.15) ◽  
pp. 76
Author(s):  
Koh Siew Khew ◽  
Chin Ching Herny ◽  
Tan Yi Fei ◽  
Pooi Ah Hin ◽  
Goh Yong Kheng ◽  
...  
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Cumulative Distribution ◽  
Interarrival Time ◽  
Single Server ◽  
Idle Period ◽  
Queue Length Distribution ◽  
Breakdown Rates ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained. 

scholarly journals An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

2021 ◽  
Vol 36 ◽  
pp. 04001
Author(s):  
Siew Khew Koh ◽  
Ching Herny Chin ◽  
Yi Fei Tan ◽  
Tan Ching Ng
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Interarrival Time ◽  
Negative Customers ◽  
Queue Length Distribution ◽  
Alternative Approach ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution ◽  
Stationary Probabilities

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue

1981 ◽  
Vol 13 (3) ◽  
pp. 619-630 ◽  
Author(s):  
Yukio Takahashi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Length Distribution ◽  
Waiting Time Distribution ◽  
Queue Length Distribution ◽  
Multiserver Queues ◽  
The Matrix ◽  
Rates Of Decay

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.

Some general results for many server queues

1973 ◽  
Vol 5 (01) ◽  
pp. 153-169 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Integral Equations ◽  
Waiting Time ◽  
Queue Length ◽  
Joint Distribution ◽  
Waiting Times ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Many Server Queues ◽  
The Many ◽  
Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

On the steady-state solution of the M/G/2 queue

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
General Solution ◽  
Laplace Transform ◽  
Queue Length ◽  
Joint Distribution ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
The Difference ◽  
Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

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