service completion probability, the mean number in the system increases in a convex form as well.
CASE 3: Let us consider another case where f(b) is a linear decreasing function of b, say for example a = f(b) = 1 – b, with the strict condition that 1 – b < b. In this case, we have
In this case, as b increases we see that decreases because the arrival probability is decreasing in the same rate as the service completion rate increases, i.e.
REMARK 1.3.– However, for the case where a= f(b) = 1-b, the arrival probability decreasing in a linear form of the service completion probability, the mean number in the system decreases in a strict convex form.
We summarize the results of the mean number in the system for the three cases in Table 1.1, with **** representing infeasible situations because the traffic intensity is greater than 1.
1.2.2. Service times dependent on interarrival times
The case of the Geo/Geo/1 system in which the service times depend on the interarrival times can be studied using the same idea and techniques as in section 1.2.1. If we define the arrival probability as a and the service time completion probability as b = g(a), it is straightforward to extend the ideas of that section directly. The queueing performance measures such as the mean number in the system, which we now define as E(X(a)) can be written as
and the stability condition given as a < g(a). Since the procedures will be the same, they will not be discussed in this chapter.
Table 1.1. Mean number in system for the three cases
| b | EX1 | EX2 | EX3 |
|---|---|---|---|
| 0.10 | **** | 0.1100 | **** |
| 0.20 | **** | 0.2400 | **** |
| 0.30 | **** | 0.3900 | **** |
| 0.40 | **** | 0.5600 | **** |
| 0.50 | **** | 0.7500 | **** |
| 0.60 | **** | 0.9600 | 1.2000 |
| 0.70 | 1.3153 | 1.1900 | 0.5250 |
| 0.75 | 0.7875 | 1.3125 | 0.3750 |
| 0.80 | 0.5236 | 1.4400 | 0.2667 |
| 0.85 | 0.3502 | 1.5725 | 0.1821 |
| 0.90 | 0.2168 | 1.7100 | 0.1125 |
| 0.95 | 0.1032 | 1.8525 | 0.0528 |
1.3. The PH/PH/1 case
In this section, the idea of Geo/Geo/1 system with interdependent interarrival and service times is generalized to the case of the PH/PH/1 system. However, first let us give a very brief review of the discrete PH distribution.
1.3.1. A review of discrete PH distribution
Consider a discrete time absorbing Markov chain with state space Xn, n = 0, 1, 2, ∙ ∙ ∙ , with Xn = 0, 1, 2, ∙ ∙ ∙ , N, where state 0 is an absorbing state. The transition matrix of this chain can be written as
where
There is a discrete random variable Y, which is said to have a PH distribution ( α,T) if one can write
Several well-known discrete distributions can be represented as PH distributions. Examples include the geometric distribution, the negative binomial distribution, to name just a few. In addition, most discrete distributions can be reasonably approximated by discrete PH distribution (see Mészáros et al. 2014 and references therein).
It was shown in Alfa (2004) that any discrete distribution, with finite support, can be represented by PH distribution with elapsed time or remaining time format. For example, consider the interarrival A and let
where
For the remaining time representation, the vector
1.3.2. The PH/PH/1 system
Consider a PH/PH/1 system with interarrival times and service times that are represented by (α,T) and (β, B), respectively, with means given as