Abstract
We have already seen Markov processes in which each state is represented by a vector. These Markov processes offer a convenient way to refer to a state (in fact, this was nothing more than giving names to states). Such was the case, for example, when we dealt with open and closed networks of queues, and with multiclass single-server queues. In this respect there is nothing new in this chapter where the states will be represented by two-dimensional vectors; that is, a state will be referred to by the pair of nonnegative integers (i, j). However, in an attempt to generalize some results from one-dimensional birth-and-death precesses, we will impose the following restrictions on the model:
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Notes
- 1.
This regime implies that the only option is in fact T + 1.
- 2.
We do not consider the case where T = 0 as the resulting system is that of an M/M/1 queue.
- 3.
The case where n = 0 makes the first server always idle and the second line becomes an M/M/1 queue. Thus, we ignore this option of the value for the threshold n.
- 4.
The other solution x = 1 is immaterial here.
- 5.
Note that \(A_{0} + A_{1} + A_{2}\) is a stochastic matrix. Moreover, its corresponding discrete-time Markov chain is a uniformization of the original continuous-time Markov process as defined in Sect. 8.5. In particular, it shares its limit probabilities with those of the original process.
- 6.
The derivation given here is due to Binyamin Oz.
- 7.
Note that for this not to be the case, s consecutive service stage completions prior to an arrival must take place.
- 8.
In [40] it is shown that the property that leads to a more explicit solution here is the fact that Q 0 is a rank-one matrix, i.e., a matrix that is the product of a column vector and a row vector. Of course, the Q 0 under consideration here is a special case of such a matrix: the column vector is (0, ⋯ , 0, 1)t and the row vector is (0, ⋯ , 0, λ).
- 9.
The model \(M/E_{r}/1\) is dual to the one given here in the sense that now the level can go down by one only from a single value for the phase. Nevertheless, an explicit solution for the limit probabilities can be derived (even without the need to solve for the root of a polynomial as is required here). See [40] for details.
- 10.
- 11.
Broadly speaking, we can say that the first queue evolves “independently” of the second queue. Yet, the two queues are not (statistically) independent.
- 12.
It is interesting to observe that \(\underline{\pi }_{j+1} = \underline{\pi }_{j}R_{1}\), j ≥ 1, where R 1 is a diagonal matrix with all its diagonal entries equal to w n . This does not refute the claim that the rate matrix R is the unique matrix where \(\underline{\pi }_{j+1} = \underline{\pi }_{j}R\), j ≥ 0.
- 13.
The latter reference states an expression due to Yoav Kerner.
- 14.
Part of this proof was communicated by Binyamin Oz.
- 15.
Note that Eqs. (12.55) are obeys for any A and B.
- 16.
This subsection calls for the previous knowledge of the subject of eigenvectors and of the spectral decomposition and can be skipped.
- 17.
This model was introduced in Sect. 4.7.3. What follows below is based on [10].
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Haviv, M. (2013). Two-Dimensional Markov Processes and Their Applications to 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_12
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