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We analyze a Coxian stochastic queueing model with three phases. The Kolmogorov equations of this model are constructed, and limit probabilities and the stationary probabilities of customer numbers in the system are found. The performance measures of this model are obtained and in addition the optimal order of service parameters is given with a theorem by obtaining the loss probabilities of customers in the system. That is, putting the greatest service parameter at first phase and the second greatest service parameter at second phase and the smallest service parameter at third phase makes the loss probability and means waiting time minimum. We also give the loss probability in terms of mean waiting time in the system.
is the transition probability from
*j*-
*th* phase to
phase
. In this manner while
and
this system turns into
queueing model and while
the system turns into Cox(2) queueing model. In addition, loss probabilities are graphically given in a 3D graph for corresponding system parameters and phase transient probabilities. Finally it is shown with a numeric example that this theorem holds.

Phase-type queueing models are one of the essential parts of the stochastic queueing models. There is an urgent need to construct phase-type distributions for complex representations of queueing models. The recent works being done in this field are: D. R. Cox shows how any distribution having a rational Laplace transform can be represented by a sequence of exponential phases [_{k}/C_{2}/s queueing system in [

There is not enough work on the studies of optimizing the orders of service parameters for Coxian queueing model so far. Considering this fact in this paper we analyze a Coxian stochastic queueing model with three phases, and the Kolmogorov equations of this model are constructed, limit probabilities and the stationary probabilities of customer numbers in the system are found. The performance measures of this model are obtained and in addition the optimal order of service parameters is given by a theorem by obtaining the loss probabilities of customers in the system. We also give the loss probability in terms of mean waiting time in the system. Finally it is shown with a numeric example that this theorem holds.

We have obtained stochastic equation systems of a Coxian queueing model with three servers in which the stream is Poisson with λ parameter. The service time of any customer at server i ^{ }and 1 − α_{i} be the loss probability of the system. This stochastic queueing model is illustrated in

Here

Kolmogorov differential equation for these probabilities is obtained. The probabilities of the process

We write Equation (2) as follows as

Furthermore, it is supposed that limiting distribution of

Steady-state equations for

We define

Let be the random variable that describes the number of customers in the system. The mean number of costumers:

Let W be the random variable that describes waiting time of customers in the system. Laplace transform of W

Mean waiting time in system of a customer for Cox(3) is found by formula (10)

Loss probability

Let

We can put three different service parameters to three stages in 3! different position. In this case there are 6 different loss probabilities.

The following theorem is given on minimization of loss probability.

Theorem 1. Putting the greatest service parameter at first phase and the second greatest service parameter at second phase and the smallest service parameter at third phase makes the loss probability minimum. That is,

Proof. Let’s suppose

Similarly,

Since,

we obtain

Finally,

Corollary. Since,

the minimum value which makes

In this section the loss probabilities are calculated for some values of system probabilities and

Under condition given in Theorem1, for the values

By constructing this stochastic queueing model, transient probabilities are obtained. Depending on these probabilities, mean number of customer in the system, the mean waiting time in this system by Laplace transform and the loss probability of any customer are given. It is shown by Theorem 1 that putting the greatest service parameter at first phase and the second greatest service parameter at second phase and the smallest service parameter at third phase makes the loss probability minimum. For the values

Placing service parameters to phases | Loss probabilities | ||
---|---|---|---|

Phase 1 | Phase 2 | Phase 3 | |

0 | 0 | 0.203703 | 0.203704 | 0.244444 | 0.24444444 | 0.33333333 | 0.33333333 |

0.2 | 0 | 0.242722 | 0.262436 | 0.272564 | 0.29752066 | 0.35532233 | 0.36090225 |

0.4 | 0.8 | 0.352845 | 0.358704 | 0.369425 | 0.37709835 | 0.4137837 | 0.41563943 |

0.5 | 0.5 | 0.351734 | 0.36976 | 0.365658 | 0.3893066 | 0.41479035 | 0.42053111 |

0.8 | 0.4 | 0.402855 | 0.431606 | 0.407646 | 0.44610302 | 0.44695789 | 0.45672369 |

1 | 0 | 0.366825 | 0.430464 | 0.366825 | 0.4516129 | 0.43046357 | 0.4516129 |

1 | 1 | 0.519078 | 0.519079 | 0.519079 | 0.51907894 | 0.51907894 | 0.51907894 |

for various values of

VedatSağlam,MuratSağır,ErdinçYücesoy,MüjganZobu, (2015) On Optimal Ordering of Service Parameters of a Coxian Queueing Model with Three Phases. Open Journal of Optimization,04,61-68. doi: 10.4236/ojop.2015.43008