# Modeling of a Coal Transshipment Complex Based on a Queuing Network
## 3 Mathematical Model
### Elements of the Coal Transshipment Complex  
The mathematical model of the complex operation is constructed in the form of an open queuing network (QN), which is understood as a set of a finite number \(S\) of queuing systems (hereinafter referred to as nodes). Requests come to an open QN from the external environment. It is usually considered an additional (dummy) node having an index 0. The request route is given by the routing matrix \(P=\left(P_{ij}\right)\), and the element \(P_{ij}\) is the probability of the request moving from Node \(i\) to Node \(j\) (\(i,j=\overline{0,S}\)) [44].  
The structural elements are described by QN nodes in different ways, depending on their type. The receiving yard (6 tracks) and terminal 1 (10 tracks) are modeled by _BMMAP_/\(G\)/\(6^{X}\)/0 and \({}^{*}\)/\(G^{X}\)/10/0 queues (Nodes 1 and 3, respectively), and the railway track between them is the \({}^{*}/G^{X}/1/0\) queue (Node 2). In these nodes, the channels are, in fact, the railway tracks, and requests are served in groups, the maximum size \(X\) of which is equal to the largest length of the train (71).  
Handling of the trains is lengthy and regulated but is affected by a large number of random factors. In this regard, according to the central limit theorem, their execution time can be taken as a random variable that obeys the normal distribution N(\(\mu\); \(\sigma\)) with the probability density function [8; 9]  
\[f(x,\mu,\sigma)=\ \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}.\]  
Its mean value \(\mu\) is determined by the average duration of the operations performed, and the standard deviation \(\sigma\) is determined by the possible deviation from the average.  
The model of the cargo yard is the \({}^{*}/M^{2}/2/200\) queue with a finite queue (Node 4), where the channels are car dumpers, and the queue is the thrust paths, each accommodating 100 cars. The channels serve two requests at a time.  
Two crushers are described separately in the form of two-channel QS with two places in a queue (two cars with coal), since they have different productivity (the \({}^{*}/M/2/2\) queue (Node 5)). The queue servicing discipline in all nodes is FIFO. The service process for similar production lines is generally considered Markovian [9; 19]. Therefore, we assume that the service time at Nodes 4 and 5 is described by an exponential distribution exp(\(\lambda\)) with intensity \(\lambda\) and the probability density function [8; 9]  
\[f(x,\lambda)=\begin{cases}\lambda e^{-\lambda x},&x\geq 0,\\ 0,&x<0.\end{cases}\]  
Two conveyor belts from the cargo yard to the warehouse and from the warehouse to the berth are described by \({}^{*}/D/2/2\) and \({}^{*}/D/2/0\) queues (Nodes 6 and 9, respectively). The operation of four SR is simulated by the \({}^{*}/D/4/0\) queue (Node 7). The warehouse is modeled by \({}^{*}/D/2/m\) with a queue for \(m=\) 1,200,000/70 \(\approx\) 17,143 places (Node 8). Its channels are fictitious and only necessary for transferring requests to Node 7. The nodes above handle requests one at a time, and their service time is deterministic and equal to 4200/70/60 = 1 min, since the speed of the belts is usually constant.  
The berthing cargo yard is modeled by two QSs. The first two-channel QS describes the operation of two ship-loading machines--the \({}^{*}/D/2/0\) queue (loaders, Node 10). The second two-channel QS with group servicing of requests and a queue for 114,000/70 \(\approx\) 1623 places simulates the loading of carriers and their departure from two berths (the \({}^{*}/G^{Y}/2/1623\) queue (Node 11)). The queue size corresponds to the largest hold capacity of any vessel. The average service time for a group of requests in the channel is 58.27 h (3495 min) and obeys the normal distribution. Vessels have a significant variation in hold capacity (see Table 2); therefore, we use the discrete uniform distribution U (\(a;b\)) to describe the size \(Y\) of the service group. Its parameters--\(a=673\) and \(b=1623\)--correspond to the smallest and largest hold capacity.