The interchangeability of tandem queues with heterogeneous customers and dependent service times

1992 ◽  
Vol 24 (3) ◽  
pp. 727-737 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Service Time ◽  
Finite Size ◽  
Arrival Process ◽  
Tandem Queues ◽  
Departure Process ◽  
Special Cases ◽  
Service Times ◽  
Special Case

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj. Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

The interchangeability of tandem queues with heterogeneous customers and dependent service times

1992 ◽  
Vol 24 (03) ◽  
pp. 727-737 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Service Time ◽  
Finite Size ◽  
Arrival Process ◽  
Tandem Queues ◽  
Departure Process ◽  
Special Cases ◽  
Service Times ◽  
Special Case

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj . Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi 's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

On the ordering of tandem queues with exponential servers

1986 ◽  
Vol 23 (01) ◽  
pp. 115-129 ◽  
Author(s):  
Tapani Lehtonen
Keyword(s):  
Service Time ◽  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Tandem Queues ◽  
Departure Process ◽  
Service Stations ◽  
Service Rates

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

On the ordering of tandem queues with exponential servers

1986 ◽  
Vol 23 (1) ◽  
pp. 115-129 ◽  
Author(s):  
Tapani Lehtonen
Keyword(s):  
Service Time ◽  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Tandem Queues ◽  
Departure Process ◽  
Service Stations ◽  
Service Rates

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

Stationary Poisson departure processes from non-stationary queues

1986 ◽  
Vol 23 (1) ◽  
pp. 256-260 ◽  
Author(s):  
Robert D. Foley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Distribution Functions ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Infinite Server ◽  
Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

Tandem queues with bulk arrivals, infinitely many servers and correlated service times

1997 ◽  
Vol 34 (1) ◽  
pp. 248-257 ◽  
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Service Time ◽  
Numerical Experiments ◽  
Queueing System ◽  
Tandem Queues ◽  
Renewal Equations ◽  
Occupancy Distribution ◽  
Service Times ◽  
Binomial Moments ◽  
Counterintuitive Result

A system of GIx/G/∞ queues in tandem is considered where the service times of a customer are correlated but the service time vectors for customers are independently and identically distributed. It is shown that the binomial moments of the joint occupancy distribution can be generated by a sequence of renewal equations. The distribution of the joint occupancy level is then expressed in terms of the binomial moments. Numerical experiments for a two-station tandem queueing system demonstrate a somewhat counterintuitive result that the transient covariance of the joint occupancy level decreases as the covariance of the service times increases. It is also shown that the analysis is valid for a network of GIx/SM/∞ queues.

Bounds for the expected delays in some tandem queues

1980 ◽  
Vol 17 (03) ◽  
pp. 831-838 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Upper Bound ◽  
Renewal Process ◽  
Arrival Process ◽  
Tandem Queues ◽  
Service Distribution ◽  
Deterministic Service ◽  
Service Times ◽  
Constant Service Times ◽  
General Service

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.

Tandem queues with bulk arrivals, infinitely many servers and correlated service times

1997 ◽  
Vol 34 (01) ◽  
pp. 248-257
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Service Time ◽  
Numerical Experiments ◽  
Queueing System ◽  
Tandem Queues ◽  
Renewal Equations ◽  
Occupancy Distribution ◽  
Service Times ◽  
Binomial Moments ◽  
Counterintuitive Result

A system of GIx /G/∞ queues in tandem is considered where the service times of a customer are correlated but the service time vectors for customers are independently and identically distributed. It is shown that the binomial moments of the joint occupancy distribution can be generated by a sequence of renewal equations. The distribution of the joint occupancy level is then expressed in terms of the binomial moments. Numerical experiments for a two-station tandem queueing system demonstrate a somewhat counterintuitive result that the transient covariance of the joint occupancy level decreases as the covariance of the service times increases. It is also shown that the analysis is valid for a network of GIx/SM/∞ queues.

Stationary Poisson departure processes from non-stationary queues

1986 ◽  
Vol 23 (01) ◽  
pp. 256-260 ◽  
Author(s):  
Robert D. Foley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Distribution Functions ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Infinite Server ◽  
Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt /∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (04) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (4) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Sign in / Sign up

Export Citation Format

Share Document