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.

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.

A note on the queueing system GI/G/∞

1968 ◽  
Vol 64 (2) ◽  
pp. 477-479 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Queueing System ◽  
Expected Value ◽  
Arrival Times ◽  
Service Times

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

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.

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.

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.

scholarly journals Comments on a Two Queue Network

2017 ◽  
Vol 6 (6) ◽  
pp. 43
Author(s):  
Samantha Morin ◽  
Myron Hlynka ◽  
Shan Xu
Keyword(s):  
Service Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Time To Completion ◽  
Time Parameters ◽  
Service Times ◽  
Interarrival Times

A special customer must complete service from two servers, each with an $M/M/1$ queueing system. It is assumed that the two queueing systems have initiial numbers of customers $a$ and $b$ at the instant when the special customer arrives, and subsequent interarrival times and service times are independent. We find the expected total time (ETT) for the special customer to complete service. We show that even if the interarrival and service time parameters of two queues are identical, there exist examples (specific values of the parameters and initial lengths a and b) for which the special customer surprisingly has a lower expected total time to completion by joining the longer queue first rather than the shorter one.

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.

scholarly journals The single server queueing system with Non-recurrent input-process and Erlang service time

1963 ◽  
Vol 3 (2) ◽  
pp. 220-236 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Service Time ◽  
Queueing System ◽  
Single Server ◽  
Input Process ◽  
Service Times ◽  
Image Position

We consider a single server queueing system in which customers arrive at the instants t0, t1, …, tm, …. We write τm = tm+1 − tm, m ≧ 0. There is a single server with distribution of service times B(x) given by where k is an integer not less than unity.

scholarly journals A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks

1999 ◽  
Vol 12 (1) ◽  
pp. 35-62 ◽  
Author(s):  
Doo Il Choi ◽  
Charles Knessl ◽  
Charles Tier
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Queueing System ◽  
Arrival Process ◽  
Tail Probabilities ◽  
Asymptotic Results ◽  
Time Service ◽  
Service Times ◽  
General Service ◽  
Stationary Probabilities

A queueing system (M/G1,G2/1/K) is considered in which the service time of a customer entering service depends on whether the queue length, N(t), is above or below a threshold L. The arrival process is Poisson, and the general service times S1 and S2 depend on whether the queue length at the time service is initiated is <L or ≥L, respectively. Balance equations are given for the stationary probabilities of the Markov process (N(t),X(t)), where X(t) is the remaining service time of the customer currently in service. Exact solutions for the stationary probabilities are constructed for both infinite and finite capacity systems. Asymptotic approximations of the solutions are given, which yield simple formulas for performance measures such as loss rates and tail probabilities. The numerical accuracy of the asymptotic results is tested.

The dependence of sojourn times on service times in tandem queues

1984 ◽  
Vol 21 (3) ◽  
pp. 661-667 ◽  
Author(s):  
Xi-Ren Cao
Keyword(s):  
Sojourn Time ◽  
Past History ◽  
Interarrival Time ◽  
Conditional Probabilities ◽  
Tandem Queues ◽  
Sojourn Times ◽  
Distributed Service ◽  
Service Times

In this paper we study a series of servers with exponentially distributed service times. We find that the sojourn time of a customer at any server depends on the customer's past history only through the customer's interarrival time to that server. A method of calculating the conditional probabilities of sojourn times is developed.

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