Talk:Jackson network

Definition of a Jackson Network

The current version of the article says: "A Jackson network consists of a number of nodes, where each node represents a queue in which the service rate can be both node-dependent and state-dependent. Jobs travel among the nodes following a fixed routing matrix."

When we say a Jackson network it is commonly understood that the service rate is not state dependent. See, for example, Stochastic processes: theory and methods by DN Shanbhag, CR Rao (page 313, Definition 2.5) or Elements of Queueing Theory by Bacelli Bremaud, (page 180, "...Jackson-type networks where the service times and routing decisions are sequences attached to nodes and not to the customers..."). One also encounters the definition given in the current version of the article in the literature (see for example, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization By Hong Chen, David D. Yao, Chapter 2).

The "jackson network" defined in the current version of the article is the class of processes studied in a 1963 paper by Jackson. The 1957 paper by Jackson that considers the classical Jackson networks (i.e., constant service rate for each node) is an earlier and more basic reference.

Based on these, perhaps the intro can be edited to include something like this:

A Jackson network consists of a number of nodes, where each node represents a queue where customers wait for service. Inter arrival times, and service times at each node are assumed to be exponentially distributed and independent and depend only on the node. When the service of a customer finishes at a node it either goes to another node or leaves the system; this choice depends only on the current node of the customer and its probabilities when written as a matrix (each row representing a node in the system) is called the routing matrix of the network. James Jackson's 1957 paper "Networks of Waiting Lines" is the groundbreaking paper on this model and computes the stationary distribution of the population size at each node when the system is in steady state. [Explain the product form of the stationary distribution].

Jackson later showed (1963, Jobshop-like Queueing Systems) that a generalization of the above model where the service rate depends on the number of customers at each node, also has a product form stationary distribution. Some authors also refer to this model as a Jackson network (see, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization By Hong Chen, David D. Yao, Chapter 2).

Aprobabilist (talk) 12:43, 15 December 2014 (UTC)

old

M/M/1 is a perfectly fine notation for a queueing system which only has one server, although it can be generalized into the category of M/M/k, the uniqueness of the sole server is usually studied as a special case, and many applications of this case also makes it important. So is M/G/1 or G/M/1.

M/M/?

An older revision suggests that the correct queuing model is M/M/s, not M/M/1, where s is the number of servers. This would make sense, though I cannot lay my hands on evidence to verify this. -- Cameron Dewe 23:36, 29 July 2006 (UTC)

Where is the definition?

In the section titled Definition, where is the actual definition? Michael Hardy (talk) 16:36, 19 February 2014 (UTC)