Conference publications

Abstracts

XVI conference

Modeling retrial queueing system with Erlang distribution of orbit's cycle time

Pustova S.V.

K.Komarova, 1, NAU IKT FKS KKSU, Kyiv, 03148, Ukraine

The multi-channel retrial queueing system with nonexponential distributive law of the sojourn time of the orbit’s cycle is considered.

In works [1] – [2] the exponential distributive law of the sojourn time of the orbit’s cycle was considered. The new seventh position in notation of Kendall’s classification of queueing systems [3] is introduced, indicating a distributive law of orbit cycle time.

Consider queueing system of $M/M/c/0/\infty //E_2 $ type, in which calls arrive in Poisson process with rate $\lambda $, and the service times have exponential distribution with the rate $\mu $. The flow of retrial calls is a two-phase Erlang process with parameter $\nu $ (density of distribution is $d(x) = (2\nu )^2 xe^{ - (2\nu )x} $).

The system functioning can be described by means of three-dimensional process $(X(t),Y(t),Z(t))$, where $X(t)$ is the number of busy channels, $Y(t)$ is the number of retrial calls on the first orbit’s phase and $Z(t)$ on the second phase at time $t$. The process $(X(t),Y(t),Z(t))$ is defined on the state space $S = \{ 0,1,...,c\} \times \{ 0,1,...\} \times \{ 0,1,...\} $.

Let $p_{kij} (t) = P\{ X(t) = k,Y(t) = i,Z(t) = j\} $ be the probability of that the system is in state $(k,i,j) $. Then the system’s analytical model in the steady state is ($p_{kij} = 0 $, if $\forall k,\forall i,\forall j < 0 $)

$(\lambda + j \cdot 2\nu + i \cdot 2\nu )p_{0ij} = \mu p_{1ij} + (i + 1)2\nu p_{0,i + 1,j - 1} $, $i \ge 0,j \ge 0 $,

$(\lambda + i \cdot 2\nu + \mu )p_{1ij} = \lambda p_{0ij} + \lambda p_{1,i - 1,j} + (j + 1)2\nu p_{0,i,j + 1} + (i + 1)2\nu p_{1,i + 1,j - 1} $, $i \ge 0,j \ge 0 $,

$\sum\lim_{k = 0}^c {\sum\lim_{i = 0}^\infty {\sum\lim_{j = 0}^\infty {p_{kij} } } } = 1 $.

References

1. Pustova S. Modeling call center operation with taking into account repeated attempts of subscribers. // Вісник НАУ, Vol. 29, No. 3, 2006. Стр. 21-24.

2. Koba E.V., Pustovaya S.V. Call Center as Retrial Queueing System. // J. of Automation and Information Sciences, Vol. 38, No. 5, 2007. Pp. 37-47.

3. Коба О.В. Системи обслуговування заявок при детермінованому часі перебування на орбіті // Вісник НАУ, 3, 2002. Стр. 69-73.

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