On sojourn time in Jackson networks of queues

1987 ◽  
Vol 24 (2) ◽  
pp. 495-510 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Sojourn Time ◽  
Network Flow ◽  
External Input ◽  
Single Server ◽  
Jackson Networks ◽  
Flow Equations ◽  
Markovian Queue ◽  
The Laplace Transform ◽  
Time Variables ◽  
Networks Of Queues

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.

On sojourn time in Jackson networks of queues

1987 ◽  
Vol 24 (02) ◽  
pp. 495-510 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Sojourn Time ◽  
Network Flow ◽  
External Input ◽  
Single Server ◽  
Jackson Networks ◽  
Flow Equations ◽  
Markovian Queue ◽  
The Laplace Transform ◽  
Time Variables ◽  
Networks Of Queues

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.

Characterizations of Poisson traffic streams in Jackson queueing networks

1979 ◽  
Vol 11 (02) ◽  
pp. 422-438 ◽  
Author(s):  
Benjamin Melamed
Keyword(s):  
Queueing Networks ◽  
Simple Graph ◽  
Single Server ◽  
Jackson Network ◽  
Jackson Networks ◽  
Equilibrium Behavior ◽  
Poisson Arrivals ◽  
Stochastic Properties ◽  
Networks Of Queues ◽  
Traffic Processes

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues. A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

scholarly journals Sojourn time distributions in a Markovian G-queue with batch arrival and batch removal

1999 ◽  
Vol 12 (4) ◽  
pp. 339-356 ◽  
Author(s):  
Yang Woo Shin
Keyword(s):  
Markov Chains ◽  
Sojourn Time ◽  
First Passage Time ◽  
Passage Time ◽  
Batch Arrival ◽  
Single Server ◽  
Negative Customers ◽  
First Passage ◽  
Markovian Queue ◽  
Sojourn Time Distributions

We consider a single server Markovian queue with two types of customers; positive and negative, where positive customers arrive in batches and arrivals of negative customers remove positive customers in batches. Only positive customers form a queue and negative customers just reduce the system congestion by removing positive ones upon their arrivals. We derive the LSTs of sojourn time distributions for a single server Markovian queue with positive customers and negative customers by using the first passage time arguments for Markov chains.

On stability of queueing networks with job deadlines

2003 ◽  
Vol 40 (2) ◽  
pp. 293-304 ◽  
Author(s):  
Amy R. Ward ◽  
Nicholas Bambos
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Single Server ◽  
Single Server Queue ◽  
Interesting Phenomenon ◽  
Single Node ◽  
Service Disciplines ◽  
Server Queue ◽  
Weakly Stable ◽  
Networks Of Queues

In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.

Characterizations of Poisson traffic streams in Jackson queueing networks

1979 ◽  
Vol 11 (2) ◽  
pp. 422-438 ◽  
Author(s):  
Benjamin Melamed
Keyword(s):  
Queueing Networks ◽  
Simple Graph ◽  
Single Server ◽  
Jackson Network ◽  
Jackson Networks ◽  
Equilibrium Behavior ◽  
Poisson Arrivals ◽  
Stochastic Properties ◽  
Networks Of Queues ◽  
Traffic Processes

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues.A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

On stability of queueing networks with job deadlines

2003 ◽  
Vol 40 (02) ◽  
pp. 293-304
Author(s):  
Amy R. Ward ◽  
Nicholas Bambos
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Single Server ◽  
Single Server Queue ◽  
Interesting Phenomenon ◽  
Single Node ◽  
Service Disciplines ◽  
Server Queue ◽  
Weakly Stable ◽  
Networks Of Queues

In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.

scholarly journals On network flow equations and splitting formulas for Sojourn times in queueing networks

1991 ◽  
Vol 4 (2) ◽  
pp. 111-116
Author(s):  
Hans Daduna
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Network Flow ◽  
Flow Equations ◽  
Sojourn Times ◽  
Recursive Evaluation

LEMOINE's network flow equations are generalized to the case of multiserver networks. These equations provide a basis for recursive evaluation of residual conditional sojourn time moments.

Time-dependent single server Markovian queue with catastrophe

2015 ◽  
Vol 9 ◽  
pp. 3275-3283
Author(s):  
R. Sudhesh ◽  
A. Vaithiyanathan
Keyword(s):  
Time Dependent ◽  
Single Server ◽  
Markovian Queue

scholarly journals Weak Convergence Limits for Closed Cyclic Networks of Queues with Multiple Bottleneck Nodes

2012 ◽  
Vol 49 (1) ◽  
pp. 60-83
Author(s):  
Ole Stenzel ◽  
Hans Daduna
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Time Distribution ◽  
Idle Time ◽  
Single Server ◽  
Sojourn Time Distribution ◽  
Cyclic Networks ◽  
Number Of Customers ◽  
Networks Of Queues ◽  
Asymptotic Behaviours

We consider a sequence of cycles of exponential single-server nodes, where the number of nodes is fixed and the number of customers grows unboundedly. We prove a central limit theorem for the cycle time distribution. We investigate the idle time structure of the bottleneck nodes and the joint sojourn time distribution that a test customer observes at the nonbottleneck nodes during a cycle. Furthermore, we study the filling behaviour of the bottleneck nodes, and show that the single bottleneck and multiple bottleneck cases lead to different asymptotic behaviours.

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

1990 ◽  
Vol 27 (02) ◽  
pp. 465-468 ◽  
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)).

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