The dependence of sojourn times on service times in tandem queues

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.

Download Full-text

The overall sojourn time in tandem queues with identical successive service times and renewal input

Stochastic Processes and their Applications ◽

10.1016/0304-4149(94)90106-6 ◽

1994 ◽

Vol 52 (1) ◽

pp. 165-178 ◽

Cited By ~ 4

Author(s):

Pierre Le Gall

Keyword(s):

Sojourn Time ◽

Tandem Queues ◽

Service Times

Download Full-text

On the independence of sojourn times in tandem queues

Advances in Applied Probability ◽

10.2307/1427176 ◽

1989 ◽

Vol 21 (2) ◽

pp. 488-489 ◽

Cited By ~ 1

Author(s):

Thomas M. Chen

Keyword(s):

Tandem Queues ◽

Sojourn Times ◽

Deterministic Service ◽

Service Times

Reich (1957) proved that the sojourn times in two tandem queues are independent when the first queue is M/M /1 and the second has exponential service times. When service times in the first queue are not exponential, it has been generally expected that the sojourn times are not independent. A proof for the case of deterministic service times in the first queue is offered here.

Download Full-text

The stationary local sojourn time in single server tandem queues with renewal input

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953399000349 ◽

1999 ◽

Vol 12 (4) ◽

pp. 417-428

Author(s):

Pierre Le Gall

Keyword(s):

Sojourn Time ◽

Stationary Regime ◽

General Distribution ◽

Single Server ◽

Tandem Queues ◽

Tandem Queue ◽

Service Times

We start from an earlier paper evaluating the overall sojourn time to derive the local sojourn time in stationary regime, in a single server tandem queue of (m+1) stages with renewal input. The successive service times of a customer may or may not be mutually dependent, and are governed by a general distribution which may be different at each sage.

Download Full-text

SOJOURN TIMES IN THE M/G/1 FB QUEUE WITH LIGHT-TAILED SERVICE TIMES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964805050205 ◽

2005 ◽

Vol 19 (3) ◽

pp. 351-361 ◽

Cited By ~ 10

Author(s):

M. Mandjes ◽

M. Nuyens

Keyword(s):

Decay Rate ◽

Sojourn Time ◽

Busy Period ◽

Service Discipline ◽

Asymptotic Decay ◽

Sojourn Times ◽

Service Times ◽

Asymptotic Decay Rate

The asymptotic decay rate of the sojourn time of a customer in the stationary M/G/1 queue under the foreground–background (FB) service discipline is studied. The FB discipline gives service to those customers that have received the least service so far. We prove that for light-tailed service times, the decay rate of the sojourn time is equal to the decay rate of the busy period. It is shown that FB minimizes the decay rate in the class of work-conserving disciplines.

Download Full-text

Single server tandem queues and queueing networks with non-correlated successive service times

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953301000338 ◽

2001 ◽

Vol 14 (4) ◽

pp. 381-398

Author(s):

Pierre Le Gall

Keyword(s):

Sojourn Time ◽

Queueing Networks ◽

Interarrival Time ◽

Single Server ◽

Tandem Queue ◽

Traditional Concept ◽

Queueing Delay ◽

Service Times ◽

Two Stages ◽

Traffic Source

To evaluate the local actual queueing delay in general single server queueing networks with non-correlated successive service times for the same customer, we start from a recent work using the tandem queue effect, when two successive local arrivals are not separated by premature departures. In that case, two assumptions were made: busy periods not broken up, and there are limited variations for successive service times. These assumptions are given up after having crossed two stages. The local arrivals become indistinguishable for the sojourn time inside a given busy period. It is then proved that the local sojourn time of this tandem queue effect may be considered as the sum of two components: the first (independent of the local interarrival time) corresponding to the case where upstream, successive service times are supposed to be identical to the local service time, and the second (negligible after having crossed 2 or 3 stages) depending on local interarrival times increasing because of broken up busy periods. The consequence is the possible occurrence of the agglutination phenomenon of indistinguishable customers in the buffers (when there are limited premature departures), due to a stronger impact of long service times upon the local actual queueing delay, which is not consistent with the traditional concept of local traffic source only generating distinguishable customers.

Download Full-text

On the independence of sojourn times in tandem queues

Advances in Applied Probability ◽

10.1017/s000186780001870x ◽

1989 ◽

Vol 21 (02) ◽

pp. 488-489

Author(s):

Thomas M. Chen

Keyword(s):

Tandem Queues ◽

Sojourn Times ◽

Deterministic Service ◽

Service Times

Reich (1957) proved that the sojourn times in two tandem queues are independent when the first queue is M/M /1 and the second has exponential service times. When service times in the first queue are not exponential, it has been generally expected that the sojourn times are not independent. A proof for the case of deterministic service times in the first queue is offered here.

Download Full-text

Sojourn times and the overtaking condition in Jacksonian networks

Advances in Applied Probability ◽

10.1017/s0001867800020218 ◽

1980 ◽

Vol 12 (04) ◽

pp. 1000-1018 ◽

Cited By ~ 10

Author(s):

J. Walrand ◽

P. Varaiya

Keyword(s):

Sojourn Time ◽

Sojourn Times ◽

Mutually Independent

Consider an open multiclass Jacksonian network in equilibrium and a path such that a customer travelling along it cannot be overtaken directly by a subsequent arrival or by the effects of subsequent arrivals. Then the sojourn times of this customer in the nodes constituting the path are all mutually independent and so the total sojourn time is easily calculated. Two examples are given to suggest that the non-overtaking condition may be necessary to ensure independence when there is a single customer class.

Download Full-text

Cumulative constrained sojourn times in semi-Markov processes with an application to pensionable service

Journal of Applied Probability ◽

10.2307/3213116 ◽

1978 ◽

Vol 15 (3) ◽

pp. 531-542 ◽

Cited By ~ 4

Author(s):

Izzet Sahin

Keyword(s):

Markov Processes ◽

Sojourn Time ◽

Working Life ◽

Partial Coverage ◽

Sojourn Times ◽

Semi Markov Processes

This paper is concerned with the characterization of the cumulative pensionable service over an individual's working life that is made up of random lengths of service in different employments in a given industry, under partial coverage, transferability, and a uniform vesting rule. This characterization uses some results that are developed in the paper involving a functional and cumulative constrained sojourn times (constrained in the sense that if a sojourn time is less than a given constant it is not counted) in semi-Markov processes.

Download Full-text

Convexity results for single-server queues and for multiserver queues with constant service times

Journal of Applied Probability ◽

10.1017/s0021900200038948 ◽

1990 ◽

Vol 27 (02) ◽

pp. 465-468 ◽

Cited By ~ 1

Author(s):

Arie Harel

Keyword(s):

Waiting Time ◽

Service Time ◽

Sojourn Time ◽

Interarrival Time ◽

Service Rate ◽

Single Server ◽

Multiserver Queues ◽

Service Rates ◽

Constant Service Times ◽

Number Of Customers

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates. These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Download Full-text