The synchronization of Poisson processes and queueing networks with service and synchronization nodes

2000 ◽  
Vol 32 (03) ◽  
pp. 824-843 ◽  
Author(s):  
Balaji Prabhakar ◽  
Nicholas Bambos ◽  
T. S. Mountford
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Poisson Processes ◽  
Finite Capacity ◽  
Random Input ◽  
Jackson Network ◽  
Limiting Behavior ◽  
Network Output ◽  
Poisson Arrivals ◽  
Buffer Capacities

This paper investigates the dynamics of a synchronization node in isolation, and of networks of service and synchronization nodes. A synchronization node consists of M infinite capacity buffers, where tokens arriving on M distinct random input flows are stored (there is one buffer for each flow). Tokens are held in the buffers until one is available from each flow. When this occurs, a token is drawn from each buffer to form a group-token, which is instantaneously released as a synchronized departure. Under independent Poisson inputs, the output of a synchronization node is shown to converge weakly (and in certain cases strongly) to a Poisson process with rate equal to the minimum rate of the input flows. Hence synchronization preserves the Poisson property, as do superposition, Bernoulli sampling and M/M/1 queueing operations. We then consider networks of synchronization and exponential server nodes with Bernoulli routeing and exogenous Poisson arrivals, extending the standard Jackson network model to include synchronization nodes. It is shown that if the synchronization skeleton of the network is acyclic (i.e. no token visits any synchronization node twice although it may visit a service node repeatedly), then the distribution of the joint queue-length process of only the service nodes is product form (under standard stability conditions) and easily computable. Moreover, the network output flows converge weakly to Poisson processes. Finally, certain results for networks with finite capacity buffers are presented, and the limiting behavior of such networks as the buffer capacities become large is studied.

The synchronization of Poisson processes and queueing networks with service and synchronization nodes

2000 ◽  
Vol 32 (3) ◽  
pp. 824-843 ◽  
Author(s):  
Balaji Prabhakar ◽  
Nicholas Bambos ◽  
T. S. Mountford
Keyword(s):  
Queueing Networks ◽  
Queue Length ◽  
Poisson Processes ◽  
Finite Capacity ◽  
Random Input ◽  
Jackson Network ◽  
Limiting Behavior ◽  
Network Output ◽  
Poisson Arrivals ◽  
Buffer Capacities

This paper investigates the dynamics of a synchronization node in isolation, and of networks of service and synchronization nodes. A synchronization node consists of M infinite capacity buffers, where tokens arriving on M distinct random input flows are stored (there is one buffer for each flow). Tokens are held in the buffers until one is available from each flow. When this occurs, a token is drawn from each buffer to form a group-token, which is instantaneously released as a synchronized departure. Under independent Poisson inputs, the output of a synchronization node is shown to converge weakly (and in certain cases strongly) to a Poisson process with rate equal to the minimum rate of the input flows. Hence synchronization preserves the Poisson property, as do superposition, Bernoulli sampling and M/M/1 queueing operations. We then consider networks of synchronization and exponential server nodes with Bernoulli routeing and exogenous Poisson arrivals, extending the standard Jackson network model to include synchronization nodes. It is shown that if the synchronization skeleton of the network is acyclic (i.e. no token visits any synchronization node twice although it may visit a service node repeatedly), then the distribution of the joint queue-length process of only the service nodes is product form (under standard stability conditions) and easily computable. Moreover, the network output flows converge weakly to Poisson processes. Finally, certain results for networks with finite capacity buffers are presented, and the limiting behavior of such networks as the buffer capacities become large is studied.

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.

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.

scholarly journals Performance Evaluation of Cloud Data Centers with Batch Task Arrivals

2013 ◽  
pp. 199-223 ◽  
Author(s):  
Hamzeh Khazaei ◽  
Jelena Mišić ◽  
Vojislav B. Mišić
Keyword(s):  
Performance Evaluation ◽  
Queue Length ◽  
Blocking Probability ◽  
General Distribution ◽  
Cloud Data ◽  
Wide Range ◽  
Accurate Performance ◽  
Cloud Data Centers ◽  
Poisson Arrivals

Accurate performance evaluation of cloud computing resources is a necessary prerequisite for ensuring that Quality of Service (QoS) parameters remain within agreed limits. In this chapter, the authors consider cloud centers with Poisson arrivals of batch task requests under total rejection policy; task service times are assumed to follow a general distribution. They describe a new approximate analytical model for performance evaluation of such systems and show that important performance indicators such as mean request response time, waiting time in the queue, queue length, blocking probability, probability of immediate service, and probability distribution of the number of tasks in the system can be obtained in a wide range of input parameters.

Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (03) ◽  
pp. 708-738 ◽  
Author(s):  
X. R. Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Jackson Network ◽  
The Mean ◽  
Service Times ◽  
A New Technique ◽  
Number Of Customers

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (1) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

scholarly journals On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

2005 ◽  
Vol 42 (01) ◽  
pp. 199-222 ◽  
Author(s):  
Yutaka Sakuma ◽  
Masakiyo Miyazawa
Keyword(s):  
Stability Condition ◽  
Queue Length ◽  
Length Distribution ◽  
Buffer Size ◽  
Finite Buffer ◽  
Jackson Network ◽  
Numerical Examples ◽  
Queue Length Distribution ◽  
Special Cases ◽  
The Stability

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

A single server tandem queue

1971 ◽  
Vol 8 (1) ◽  
pp. 95-109 ◽  
Author(s):  
Sreekantan S. Nair
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Single Server ◽  
Queueing Model ◽  
Tandem Queue ◽  
Limiting Behavior ◽  
Virtual Waiting Time ◽  
Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem.Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (03) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

Sign in / Sign up

Export Citation Format

Share Document