Year: 2023
Type: conference-paper
Abstract: In the presented article, we consider the queuing model GI|G|1|∞ with the FIFO service discipline, and the service time of the n-th call is denoted by , n≥ .It is assumed that the sequences of RV's, where = -, n≥ and { }, n≥1 are sequences of independent, identically distributed RV's with DF A(t), A(+0)=0 and B(t), B(+0)=0, respectively.The queuing model is investigated under the condition of model loading = <1, where =E and =E for all n≥ and IHR DF B(x), which is equivalent to not increasing by the relation (x+t)/ (t), where we denote IHR DF (t)= 1-B(t) for t∈[0, + ∞).Customs are numbered in the order of receipt by the numbers 1,2, ..., i.e.At the moment t=0, there are no calls in the system.Denote by , ≥ the waiting time until the service beginning ofcustomer and by w(t), t>the virtual waiting time starting at the moment t, or more precisely: the length of the time interval starting at the moment t and ending at the time when the system is free from customers that arrived in the system before time t.It is known that the following stationary limits exist: ⇒ , → +∞ and !⇒ * , → +∞, (1) where ⇒ denotes the sign of weak convergence (see [4], p.68 and p. 139).Denote = # and = #$ , where E is a sign of mathematical expectation, and assume that 0< <+∞ and 0< <+∞.Due to Theorem 2.2 (see [4], p.73) and Theorem 3.2 (see [4], p.139), it follows that under the condition, = <1 ( is the traffic intensity of the system) (2) the DF's % &! = ' ≤ &! and % * &! = ' * ≤ &!, )*+ ,*-,+* ./0,i.e., % +∞! = % * +∞! = .In the present paper, the queueing model GI|G|1| ∞ in stationary conditions is considered.For waiting time 1234 56 % &!, ≥ , and the virtual waiting time at the moment t !exist ⇒ , → +∞ and !⇒ * , → +∞, where ⇒ denotes the sign of weak convergence.In the paper, the model with FIFO discipline and = <1 (see (2)) and DF B(x) of customers service times is IHR DF.Then both random variables w and w* are IHR.Keywords-increasing hazard rate (IHR), the k-th ladder point, the k-th ladder height, virtual waiting time, Takac's, Cohen's and Hooke's formulas.
DOI: 10.51408/csit2023_42
Author: Robert Chitchyan
Language: English
Field: Decision Sciences
Domain: Social Sciences
Open Access status: bronze
OpenAlex ID: W4386743263