href="#fb3_img_img_9ae9cb60-4353-5e9b-82ad-bff51297dc96.jpg" alt="image"/> where
where α = [α(1), α(2), ∙∙∙ , α(k)] and the operator * is the Khatri-Rao (K-R) product (Khatri and Rao 1968) , and here we have
In applying K-R operator, here the first vector ψ is partitioned into K, 1 × 1 vectors, i.e. into scalars in order to properly apply the K-R operator.
This is better illustrated with an example.
Example
Consider a case of a service time S with
Then we have a system with service time distribution vector given as s = [s1, s2, s3, s4] and interarrival time distribution given as a PH with
Numerical examples
Let s =[0.1, 0.3, 0.4, 0.2] and
suppose we select three mappings of s with
but different
Let Aj be the interarrival times for the jth mapping and
Table 1.2. Complementary Cumulative distributions of interarrival times
| k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| P1(k) | 0.6490 | 0.4886 | 0.3311 | 0.2382 | 0.1704 | 0.1246 | 0.0919 | 0.688 | 0.0522 |
| P2(k) | 0.7110 | 0.5526 | 0.4225 | 0.3361 | 0.2705 | 0.2210 | 0.1823 | 0.1516 | 0.1268 |
| P3(k) | 0.6740 | 0.5128 | 0.3657 | 0.2751 | 0.2084 | 0.1613 | 0.1265 | 0.1006 | 0.0810 |
In future works, we study how the different arrangements of s affect the features and distribution and moments of the interarrival times. This can be used to show how one can control the arrival process and hence the queue performance by using different selections of the vector s.
1.4.2. A queueing model with interarrival times dependent on service times
Consider a single-server queue with service times distribution vector, given as s of M dimension and K independent interarrival times, from which a customer’s next interarrival is selected based on the service time experienced by customers. It is clear that the true interarrival times is Markovian Arrival Process (MAP), which is constructed from the K interarrival times and the probability vector ψ of dimension K that was created by mapping service distribution as shown in section 1.4.1. The PH representation of interarrival times has κ = k1 + k2 + ... + kK, where kj is the number of phases of the jth interarrival time. We let the service times be represented by an elapsed time PH distribution with representation (β, B) of order M. Also let b = 1 – B1.
At time n = 0, 1, 2, 3, ∙∙∙, let Xn be the number of customers in the system, Yn the phase of arrival and Jn the phase of the ongoing service. The stochastic process {(Xn,Jn,Yn),n≥ 0} is a DTMC with transition matrix P given as
where
This Markov chain is a simple Quasi-Birth-Death (QBD), which can be analyzed using the matrix-analytical methods for discrete time queues (Neuts 1981; Alfa2016).
If this system is stable, which we will assume it is, then there is a unique probability vector
Further, we have
