Queueing Theory 1. Nikolaos Limnios

Queueing Theory 1 - Nikolaos Limnios


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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.

      1.2.2. Service times dependent on interarrival times

      and the stability condition given as a < g(a). Since the procedures will be the same, they will not be discussed in this chapter.

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

      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

with
for at least one i. Also define t = 1 – T1, where 1 is a column of ones.

      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

. In the elapsed time format, the matrix T has only its superdiagonal elements that are non-zero and every other element is zero, and the vector α = [1, 0, 0, ∙∙∙ , ], and the matrix T is written as

      where

      For the remaining time representation, the vector

is the vector of the discrete distribution and the matrix T is written as

      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


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