Diffusion Approximation for an Open Queueing Network with Limited Number of Customers and TimeDependent Service
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Abstract
The object of this research is an open queueing network (QN) with a single class of customers, in which the total number of customers is limited. Service parameters are dependent on time, the routing of customers is determined by an arbitrary stochastic transition probability matrix, which is also depends on time. The service times of customers in each queue of the system is exponentially distributed with FIFO service, and it is assumed that a birth and death process generates and destroys the customers. The random vector, which determines the network state, forming a Markov random process is introduced. The purpose of the research is an asymptotic analysis of this Markov process, describing the queueing network state with a large number of customers, obtained from a system of differential equations, and used to find the mean relative number of customers in the network queues at any time. The results are illustrated with a specific example. This approach can be used tp model processes of customer service in the insurance companies, banks, logistics companies and other cybereconomical or service organizations.
Keywords
Expected revenues Limited number customer Diffusion approximation Distribution density for the expected revenueIntroduction
It is well known that various real systems modelled by queueing networks, such as computer systems, networks or Cloud servers, have timedependent parameters [7, 8, 12]. In the process of their design and optimisation [9] one needs to be able to determine their time dependent probability characteristics.
Exact results for nonstationary state probabilities of Markov networks are difficult to obtain because of the high dimensionality of the systems of differencedifferential equations which they satisfy. To find these probabilities under high load conditions, the diffusion approximation method is often used [1, 2, 3, 6].
Queueing networks (QN) are mathematical models of various economical and technical systems, associated with the of service of customers, jobs or network packets [4, 5, 10]. Although the total number of customers in such systems may be limited, their total number changes over time, what allows us to uses models of open QN with limited numbers of customers.
The Derivation of the System of Differential Equations for the Average Relative Number of Customers in the Network Systems
Theorem 1
Proof

from state \(({\mathbf {k}}+I_iI_j,t)\) can get to \(({\mathbf {k}},t+ \Delta t )\) with probability \(\mu _{i} (t)\min (m_{i} (t),\) \(k_i(t)+1)p_{ij}(t) \Delta t+ o(\Delta t), i,j= \overline{0,n}\)

from state \(({\mathbf {k}}+I_0,t)\) can get to \(({\mathbf {k}},t+ \Delta t )\) with probability \(\lambda ^_0(t)(k_0(t)+1)\Delta t+ o(\Delta t),\)

from state \(({\mathbf {k}}I_0,t)\) can get to \(({\mathbf {k}},t+ \Delta t )\) with probability \(\lambda _{0}^{ + } (t)\left( {K  \sum\nolimits_{{i = 0}}^{n} + 1} \right)\Delta t + o(\Delta t),\)

from state \(({\mathbf {k}},t)\) can get to \(({\mathbf {k}},t+ \Delta t )\) with probability
\(1  \left[ {\sum\nolimits_{{i = 0}}^{n} {\mu _{i} } (t)\min (m_{i} (t),k_{i} (t))p_{{ij}} (t) + \lambda _{0}^{  } (t)(k_{0} (t) + 1) + \lambda _{0}^{ + } (t)(K  \sum\nolimits_{{i = 0}}^{n} {k_{i} } (t) + 1)} \right]\Delta t + o(\Delta t)\),
and from other states with probability \(o(\Delta t)\).
Example
Conclusions
In this paper, Markov queueing network with a limited number of the same type of customers has been investigated. The number of customers in the system varies in accordance with the birth and death in process. To obtain a system of differential equations for the average number of customers in the system, the method of diffusion approximation is applied, allowing one to find the average number with high accuracy for a large number of customers.
The results may be useful in modelling and optimization of customer service in computer systems, networks and in insurance companies, banks, logistics companies and other socioeconomic and cyberphysical organizations [8, 12].
Notes
References
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