# Open, Closed, and Mixed Networks of Queues with Different Classes of Customers

@article{Baskett1975OpenCA, title={Open, Closed, and Mixed Networks of Queues with Different Classes of Customers}, author={Forest Baskett and K. Mani Chandy and Richard R. Muntz and Fernando G. Palacios}, journal={J. ACM}, year={1975}, volume={22}, pages={248-260} }

We derive the joint equilibrium distribution of queue sizes in a network of queues containing N service centers and R classes of customers. The equilibrium state probabilities have the general form: P(S) - Cd(S) $f_1$($x_1$)$f_2$($x_2$)...$f_N$($x_N$) where S is the state of the system, $x_i$ is the configuration of customers at the ith service center, d(S) is a function of the state of the model, $f_i$ is a function that depends on the type of the ith service center, and C is a normalizing…

## 2,484 Citations

OF QUEUE LENGTHS AND WAITING TIMES IN CLOSED, PRODUCT-FORM QUEUEING NETWORKS

- Mathematics
- 1989

Little's theorem states that under very general conditions L = 1 W, where L is the time average number in the system, W is the expected sojourn time in the system, and A is the mean arrival rate to…

Asymptotic analysis of closed queueing networks and its implications to achievable service levels

- MathematicsPERV
- 2010

An asymptotic analysis of the underlying closed queueing network is presented by driving the population size to infinity and studying the asymPTotic behavior of the service availability, which allows us to derive the exact order asymptonotic Behavior of many performance metrics including throughput, server utilization, mean response time, and service availability.

A generalization of Little's Law to moments of queue lengths and waiting times in closed product form queueing networks

- Mathematics
- 1988

Little's Theorem states that under very general conditions
$$L = \lambda W,$$
where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean…

On First-Come First-Served Versus Random Service Discipline in Multiclass Closed Queueing Networks

- Mathematics
- 1997

We consider multiclass closed queueing networks. For these networks, a lot of work has been devoted to characterizing and weakening the conditions under which a product-form solution is obtained for…

Networks of Queues with Customers, Signals and Arbitrary Service Time Distributions

- Computer ScienceOper. Res.
- 1995

A new definition of signals is proposed that depends on the service effort and provides a product-form solution in a network of queues with multiple classes of customers, signals, and arbitrary service time distributions.

Network of Single-Server Queues with Dependent Service Times

- Computer ScienceQueueing Syst. Theory Appl.
- 2001

A network of single-server nodes fed by customers of several classes is considered and the stationary distribution is derived, which is not a product form, but for both open and closed network, sufficient invariance conditions are found.

On Generalized Networks of Queues with Positive and Negative Arrivals

- Mathematics
- 1993

The product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the current state of the system is independent of the past departure processes.

A note on networks of infinite-server queues

- Mathematics
- 1981

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which…

Exact Solutions for Networks of Queues with Blocking-After-Service

- MathematicsTheor. Comput. Sci.
- 1994

Maximum Throughput in Finite-Capacity Open Queueing Networks with Product-Form Solutions

- Business
- 1977

A finite-capacity open queueing network with independent balance is considered, where the rate of flow into a state due to a customer entering a stage of service is equal to the flow out of that state dueto a customer leaving that stage ofservice.

## References

SHOWING 1-10 OF 33 REFERENCES

Closed Queuing Systems with Exponential Servers

- MathematicsOper. Res.
- 1967

It is found that the distribution of customers in the closed queuing system is regulated by the stage or stages with the slowest effective service rate, which means that closed systems are shown to be stochastically equivalent to open systems in which the number of customers cannot exceed N.

The dependence of computer system queues upon processing time distribution and central processor scheduling

- MathematicsOPSR
- 1972

The stationary distribution of the number of jobs being served by a processor-sharing central server is independent of both the distribution of service times and the distribution of interarrival…

Closed Finite Queuing Networks with Time Lags and with Several Classes of Units

- MathematicsOper. Res.
- 1968

The authors have previously investigated a broad class of closed, finite queuing networks with time lags, in which all units were considered to be identical. In this paper, that work is extended to…

Queueing Analysis of a Multiprogrammed Computer System Having a Multilevel Storage Hierarchy

- Computer ScienceSIAM J. Comput.
- 1973

A class of closed queueing network models which can be used to represent certain features of multiprogrammed computer systems having multilevel storage hierarchies, and derives simple expressions relating the work rates for different stages and obtains simple upper bounds on theWork rates.

Analysis of system bottlenecks using a queueing network model

- Mathematics
- 1971

A surprising result is that optimal performance is not attained when queue lengths and processor utilization percentages are equal, but rather when the fastest processor has the longest expected queue and is in effect creating a system bottleneck.

An Analysis of the Machine Interference Model

- MathematicsIBM Syst. J.
- 1971

The asymptotic properties of the machine interference model were described and it was shown that the size dependencies of these properties were of order 1/n in regions of the most interest, that is, w < 1.

Design automation and queueing networks: An interactive system for the evaluation of computer queueing models

- Computer ScienceDAC '72
- 1972

This paper presents the theory, algorithms and some programs for a system which departs radically from previous attempts at computer design aids for queueing network analysis, and allows the analyst greater freedom in choosing the models best suited for his system.

A use of complex probabilities in the theory of stochastic processes

- Mathematics
- 1955

The exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions…

I I I Network models for large - scale time * sharing systems Tech

- 1973

An analytic model of a mult iprogrammingsystem Including a job mix Rep TR4

- Dep. of Computer Sciences, U. of Texas
- 1972