Abstract
We next define the model of a discrete state-space continuous-time Markov process. Let N = { 0, 1, 2, …,n} be the possible states of a process. The value of n can be finite or infinite but N should be discrete. Note that we use a numerical value to present a state just for convenience. In fact, state-i is nothing more than a name and is selected for reference purposes. Yet, in many cases these names have a meaning and their use may lead to a natural presentation of a state, such as the number of customers in a queue (when applicable).
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Notes
- 1.
Recall that in this case we say that the arrival process is Poisson with rate λ.
- 2.
This model appears in [30], p. 23
- 3.
Appeared in [30], pp. 29–30.
- 4.
Also called Jackson networks see [28].
- 5.
e i stands for the i-th unit vector. Specifically, it is a vector full of zeros but the i-th entry comes with a one.
- 6.
By irreducibility we basically mean the same as in (3.3). Specifically, for any pair of ordered states i and j, there is a positive probability that a process currently in i reaches j sometime later. This is equivalent to assuming that the topology of the transition rates (i.e., a network such that an edge from state k to state m exists if and only if q km > 0) is such that there exists a directed path from i to j.
- 7.
Note that an empty product is defined as 1.
- 8.
State c refers here to the case where only one class-c customer is in the system.
- 9.
The derivation below appears in [22], p. 132.
- 10.
- 11.
Admittedly, in the case where μ c = μ, the total number in the system is in fact an M/M/1 process with an arrival time Σ c λ c . In particular, this total number is a Markov process of a reduced state space: all states of the type (c 1, …, c n ) with a common n collapse into a single state.
- 12.
This phenomenon is sometimes referred to as the “M ⇒ M” property.
- 13.
This question is based on [1].
- 14.
Part of this exercise is based on [47] pp. 90–91.
- 15.
The formula for π s is known as the Erlang loss or the Erlang B formula. Indeed, this is the probability that an arrival to the system finds all server busy and hence is lost.
- 16.
This is known as the Erlang C formula. See [47], p. 99.
- 17.
The results stated here appear in [22], p.132.
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Haviv, M. (2013). Continuous-Time Markov Chains and Memoryless Queues. In: Queues. International Series in Operations Research & Management Science, vol 191. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6765-6_8
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