A Survey on Queueing Systems with Mathematical Models and Applications

Queueing theory is applied in many of the daily life activities including computer networks, telecommunication systems, traffic flow systems, airport scheduling systems, banking and logistic operations and so on. Besides all these, queuing system is applied in the manufacturing industries as well. Items produced by industries have to be delivered to the retailers and then to the customers. If there is the proper chain to deliver those items, it can save time and money. Products of the industries can be delivered together in numbers but one machine can produce only one item at a time following a sequential order. Those produced items should be supplied to the wholesalers and to the retailers turn by turn maintaining a proper queue. In this sense, we can observe a close relationship between queuing system and supply chain management, which is described in rest of this Section.

Bhaskar and Lallement used the concept of supply chain to find the minimum response time for the delivery of items to the final destination through different stages of network. They identified the appropriate route of the least response time and calculated the performance measures like average queue lengths, average response times, and average waiting times of the jobs in the supply chain. They have proposed a model to calculate the mean and variance of the number of customers in the system as the follows:

$\begin{aligned} &\mathrm{E}(\mathrm{N})=\frac{1}{1-\mathrm{e}^{-(1-\rho)}} \\ &\sigma_{\mathrm{N}}^{2}=\frac{\mathrm{e}^{-(1-\rho)}}{\left(1-\mathrm{e}^{-(1-\rho)}\right)^{2}} \\ &\text { where, } \rho=\frac{\text { mean service time }}{\text { mean inter-arrival time }}=\frac{2 \lambda}{\mu(b+a)} \text { for all } \mathrm{a}, \mathrm{b} > 0 \text { and } \mathrm{b} > \mathrm{a} \text {. } \end{aligned}$

Likewise, if $R$ denotes the response time and $W$ is the waiting time in the queue, then mean response time and the mean waiting time has been expressed as

$\mathrm{E}(\mathrm{R})=\frac{1}{\lambda\left(1-\mathrm{e}^{-(1-p)}\right)}$

$E(\mathrm{~W})=\frac{\mu-\lambda+\lambda e^{-\left(1-\frac{2 \lambda}{\mu(b+a)}\right)}}{\lambda \mu\left(1-e^{-\left(1-\frac{2 \lambda}{\mu(b+a)}\right)}\right)}$

Average number of jobs found on the server has been determined by the formula $E(N)-E(Q)=\frac{\lambda \mu-\mu+\lambda-\lambda e^{-(1-p)}}{\lambda \mu\left(1-e^{-(1-p)}\right)}$

Boulaksil proposed a model to determine the safety stock levels in supply chain systems which are facing demand uncertainty. He reported that supply chain would meet a high level of customer service if large portion of the safety stocks are placed downstream. Teimoury et al. determined holding, back ordering and ordering cost function for GI/G/1 queueing model. They proposed an inventory model for batch products along with some numerical examples of manufacturing supply chain network to analyse performance evaluation. Liu et al. evaluated the performance of serial manufacturing and supply systems with inventory control by developing a multi-stage inventory queue model and a job queue decomposition approach. Then they presented an efficient procedure to minimize the overall inventory in the system maintaining the required service level. Sivakumar et al. number of customers in the pool and the inventory level where demand during stock out periods either enter a pool having finite capacity $N($ or leave the system with a predefined probability. Andriansyah et al. used generalized expansion method to evaluate predefined probability. Andriansyah et al. used generalized expansion method to evaluate simulation. Experiments for a large number of settings and different network topologies were also presented. They derived the formula for the throughput at node $i$ as

$\theta=\lambda\left(1-p_{c}\right)=\lambda\left(1-\frac{\frac{(\lambda / \mu)^{c}}{c !}}{\sum_{i=0}^{c} \frac{(\lambda / \mu)^{i}}{i !}}\right)$

where $\mathrm{P}_{c}$ is the probability of a customer being blocked for $\mathrm{M} / \mathrm{M} / \mathrm{c} / \mathrm{c}$ queueing model. Mishra and Yadav considered a clocked queueing network with renewal model and used it to develop a computational approach for the analysis of cost and profit structure in the system. They found its optimality with respect to arrival and service parameters of the system. Mary and Christina proposed the procedure to find the total average cost in terms

of crisp values for $M_{(\mathrm{m}, \mathrm{N})}^{\mathrm{X}} / \mathrm{M} / \mathrm{BD} / \mathrm{M}$ with fuzzy parameters considering many other factors with some numerical example for the validity of the proposed system. Smith used mean value analysis algorithm to study the material handling and transportation networks in finite buffer closed $\mathrm{M} / \mathrm{M} / 1 / \mathrm{K}$ queueing system. Babadi et al. applied queueing systems in nylon plastic manufacturing and recycling centres using Jackson network to minimize the average delay to deliver products, total cost and transportation cost which was checked by the sensitivity analysis by changing the parameters. Vahdani and Mohammadi proposed a bi-objective optimization model in a closed loop supply chain network in which general multi-priority and multi-server queuing system for parallel processing execution has been used to minimize the cost and maximize the profit. In order to calculate the queue waiting time of arrival products into the forward flow to the bidirectional facility, following formula has been used

$\mathrm{Wf}_{\mathrm{qb}}^{(\mathrm{p})\pm}=\frac{1}{\mathrm{Af}_{\mathrm{b}}^{\pm} \times \mathrm{Bf}_{\mathrm{b}, \mathrm{p}-1}^{\pm} \times \mathrm{Bf}_{\mathrm{b}, \mathrm{p}}^{\pm}}$for all $\mathrm{p}$

$A f_{b}^{\pm}=\left[c_{b} !\left(c_{b} \mu f_{b}-\lambda f_{p b}^{\pm}\right)\left(\dfrac{\lambda f_{p b}^{\pm}}{\mu f_{b}}\right)^{c_{b}} \sum\limits_{j=0}^{c_{b}-1}\left\{\frac{\left(\dfrac{\lambda f_{p b}^{\pm}}{\mu f_{b}}\right)^{j}}{j !}\right\}+c_{b} \mu f_{b}\right]$

for all b

and

$\mathrm{Bf}_{\mathrm{b}, \mathrm{p}}^{\pm}=1-\frac{\sum_{\mathrm{p}^{\prime}=1}^{\mathrm{p}} \lambda \mathrm{f}_{\mathrm{p}^{\prime} \mathrm{b}}^{\prime}}{\mathrm{c}_{\mathrm{b}} \mu \mathrm{f}_{\mathrm{b}}}$, with $\mathrm{Bf}_{\mathrm{b}, 0}=1$

The other notations used in the model are described as follows:

$\mathrm{Wf}_{\mathrm{qb}}^{(\mathrm{p})}\pm$ Waiting time in the queue of forward flow of product with priority $\mathrm{p}$ in bidirectional facility b;

$c_{b}=$ Number of service provider at bidirectional facility $b$;

$\mu \mathrm{f}_{\mathrm{b}}=$ Service rate at bidirectional facility b for forward products;

$\lambda f_{\mathrm{pb}}\pm$ Arrival rate of forward flow of product $p$ to bidirectional facility $b$.

Diabat et al. used queueing approach to determine the number and location of distribution centres, the assignment of retailers to distribution centres, and the size and timing of orders for each distribution centres providing some numerical results. They proposed a model in which

$\mathrm{P}_{\mathrm{K}}(0)=\frac{\lambda_{\mathrm{K}}}{\lambda_{\mathrm{K}}+\mathrm{Q}_{\mathrm{K}} \mu\left(1+\frac{\mu}{\lambda_{\mathrm{K}}}\right)^{\mathrm{S}_{\mathrm{K}}}}$

Each opened distributions centres orders to the supplier when its inventory level is less than $S+1$. Then the expected amount of recorders $\left(R_{K}\right)$ is calculated by

$\mathrm{R}_{\mathrm{K}}=\lambda_{\mathrm{K}} \mathrm{P}\left(\mathrm{S}_{\mathrm{K}}+1\right)=\mu\left(1+\frac{\mu}{\lambda_{\mathrm{K}}}\right)^{\mathrm{S}_{\mathrm{K}}} \mathrm{P}_{\mathrm{K}}(0)$

On the other hand, when the level at distribution centre located at site $\mathrm{K}$ is equal to zero and arriving demands are lost, then the expected amount of lost sales $\left(\Gamma_{K}\right)$ is $\Gamma_{K}=\lambda_{K} P_{K}(0)$

And the expected amount of inventory $\left(\mathrm{MI}_{\mathrm{K}}\right)$ has been obtained by

$\mathrm{MI}_{\mathrm{K}}=\sum_{j=0}^{\mathrm{Q}_{\mathrm{k}}+\mathrm{S}_{\mathrm{K}}} \mathrm{jP}_{\mathrm{K}}(\mathrm{j})$

where $\mathrm{Q}_{\mathrm{K}}$ and $\mathrm{S}_{\mathrm{K}}$ are the recorder quantity and recorder point at distribution centre $\mathrm{K}$. Wang et al. collected a review defining supply chain and discussing literature in the areas, namely service supply management, service demand management and the coordination of service supply chains to observe the state in each area. He derived supply risk sharing contracts for the equilibrium between the recycling price decision and the remanufacturing quantity decision using game theory illustrating some numerical examples for managerial results. Zhalechian et al. studied environmental impact of sustainable closed-loop location - routing -inventory model using a stochastic-probabilistic programming approach presenting some real-world applications. Sadjadi et al. derived optimization model using queuing approach for allocation of the retailers' demands, and inventory replenishment decisions so as to minimize the total expected cost of location, transportation and inventory. Jin formulated link transmission model and link queue model defining demand and supply to present queuing models for a point queue and their discrete versions.