Mathematical Model For Forwarding Packets In Communication Network

In recent years, SDN technology has been applied to several networks such as wide area network (WAN). IT provides many beneﬁts, such as: enhancing data transfer, promoting Application performance and reducing deployment costs. Software Deﬁned-WAN networks lack studies and references. This paper introduced a system for SD-WAN network using PH/PH/C queues. It concentrates on the study of algebraic estimates the probability distribution of the system states. The Matrix-Geometric solution procedure of a phase type distribution queue with ﬁrst-come ﬁrst-served discipline is used.

After the success of the first phase, vEdge router will communicate to the Control Plane to forward packets according to their routing table entries based on the decisions taken the SD-WAN controller system and interchange forwarding information with the SD-WAN controllers over the Overlay Management Protocol (OMP). The vSmart controller receives all routing and topology information from every vEdge router, calculates the best paths based on the policies, and then re-advertises the results to all other vEdge routers but never participate in the data-plane (packet forwarding). The operations of forwarding in packet vEdge router are arranged as shown in : Fig 4.

PROPOSED QUEUING MODEL FOR SD-WAN NETWORK
The Ph/Ph/C queue model consists of two processes as following: one is the arrival process model by Hyper-exponential Distribution with probability α i , ∑ k i=1 α i =1 and rates are given by:λ 1 ,λ 2 ,λ 3 and the other is a service process model by Hypo-exponential Distribution with two-stage hypo-exponential distribution for the service process with-its parameter µ 1 ,µ 2 as shown in Fig 3.   Figure 3. Ph/Ph/C queue with hyper-exponential and hypo-exponential

Queuing model of SD-WAN controller System
With SD-WAN under its management, N of the vEdge routers are supposed to be the packet flows with arrival rate:(λ 1 , λ 2 ,. . . , λ i ). Subject to Phase-Type distribution with probability :(α 1 , α 2 ,. . . ,α i ), the packet gets service at a rate µ i with probability α i and leave the controller. Only one packet enters the controller and then comes out after serving it.
λ c is the volume of arrivals for unit time (packet/ms), arrival rate λ i , i≥1 and arrival rate are applied to the hyper-exponential distribution by F(.) 7 . The probability density function of the arrival packet X can be given by: x otherwise while the cumulative distribution function can be expressed as: where ∑ α i = 1, α i is the probability that X will take on the form of the hyper-exponential distribution with rateλ i . The first and second moments may be expressed as 12 : The distribution F(.) on the interval [ 0,∞) can be Phase Type distribution as its representation (η, T ) where η is a row vector of dimension m denotes Probability that phase is taken and T is a square matrix of dimension m that denote arrival processes 13 . The transition matrix of arrival rateT is given as follows: The generatorT of a Markov chain can be written as:T = T T 0 0 0 and initial probability distributionὴ is given by:ὴ . Where u points a column vector with all elements equal to one, and the vector T 0 = -Tu. The coefficient of variation is:

Queuing model of vEdge router (branch)
Assuming that all packet switching process in vEdge routers are series of sequential k-phase exponential distribution. Each with its own rate µ i , the rate of the i th distribution. Which follows a hypo-exponential distribution 14 . The service process distribution H(.) can be represented by the pair (σ , S) of dimension n, where: S is service process and σ is initial probability. In this paper it is assumed that there are two phases of service inside SD-WAN controller.The transition rate matrix = service processes =S , thus we can write generatorS of such a Markov chain ? as: as well as Initial probability distributionσ as : Note that any initial probability must be equal one, so that η u = 1 and σ u = 1, where u is a unity column vector. Whereas a probability distribution H (.) 15 on the interval [0, ∞) is of phase type if it can arise as the absorption time distribution of (k + 1) state Markov chain with m transient states 1, 2,. . . , k and the absorbing state is 0. The probability density function (pdf) as can be given by: The cumulative distribution function is given by: The expectation, variance, and squared coefficient of service process may be expressed as: The coefficients of variation are always < 1.

MATHEMATICAL ANALYSIS OF PACKET FORWARDING
vEdge router analyses, the first step in the packet forwarding process is to perform a routing lookup on the destination IP of the packet in the routing table. If the match is successful, the configured action is performed usually where forwarded data to the next step, or through a specific port. Otherwise, the packets will be dropped or sent to a temporary store for processing or resent to the SD-WAN controller system. According to the policies set by the SD-WAN controller system , the action applied (by default it is reject).All packets that sent in the first and second Phase from the SD-WAN system to the vEdge routers are forwarded in a form of service queuing packets. Represented in two phases of service apply to it phase-type distribution and provided by a group of controllers as follows: The first Phase is the preparation, authentication and initial-configuration, it takes place in the orchestration plane, and it is served by two types of controllers :( Zero-Touch Provisioning (ZTP) server and vbond controller).The second Phase is the data flow between the ends of the network, forwarding packet and policies between network devices at the data plan. While the packets sent by the vEdge router are forwarded in a form of arrival queuing packets.as given below: The probability that the vEdge router is sending arrival packet to SD-WAN system asking for more information for the initial contact information, update operation system, upgrade or authentication process, probability that the vEdge router is sending arrival packet to SD-WAN system asking information about routing, forwarding data, update routing table or topology, and probability that the vEdge router is sending to arrival packet to SD-WAN system asking to Provide it with missing information and data from database which is in the redundancy controller.

Mathematical analysis:
There are four basic steps to a solving phase-type queuing systems using matrix-geometric approach as follows: 1. Building the sub-matrices: suppose that the phase arrival process is given as r a = r 3 and services process given as r s =r 2 .
The block sub-matrices A 0 , A 1 , A 2 , B 00 , B 10 , B 01 can be constructed by applying matrix-geometric approach 14 yields the following:r a =3 and r s =2 that is : It was mentioned in section 3.2 that : S, S 0 , T , T 0 ,η, σ and I n is the identity matrix of order n ,where n either (r a or r s ), A 0 represented service completions at rate µ 2 , A 2 represented arrivals completions at rateα i λ i based on number of states,

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A 1 super-diagonal elements represented service completions at rate µ 1 , and the matrices B i j represent initial conditions. The transition rate matrix (Q) given as: Since π i is vector refers to unique stationary probability distribution (steady-state distribution) can describe a continuoustime Markov chain as given by this equation: From equation 10 the general case in PH/PH/C queue can be written as: by dividing the equation to µ to produce π i+1 =( λ +µ µ )π i -( λ µ )π i−1 = ( λ µ )π i i= 1,2,. . . . knowing that the utilization factor denotes ρ = λ µ hence the final equation of stationary probability may be written as: as geometrically In equation 12 assuming that π 0 is known, hence evaluating the value of π i is easy to be obtained. In QBD process, the parameter ρ will be defined a square matrix R of order k× k and π i will be a vector of order K, where π i = (π 0 , π 1 ,π 2 ,. . . , π i ). Accordingly, equation 12 can be re-expressed as: π 0 B 00 + π 1 B 10 = 0, π 0 B 01 + π 1 A 1 + π 2 A 0 = 0 , By substituting in equation 13 can be found R as follows: 2. Form Neuts' R matrix from Equation 14is as follows: Where V=A 2 A 1 −1 , W=A 0 A 1 −1 , with initial value R (0) =0, and k=1,2,. . . .

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3. Solve the boundary equations as follows: In the system as described previously in Fig. (3) given by Q matrix, the boundary equations are: π 0 B 00 A 1 + RA 0 =0 , solved for π 0 and thereafter used π i =π 0 R i ,i=1,2,. . . , to obtain the stationary probability vector as: where π i =π 1 R i−1 ,i=2,3,. . . to provide a unique solution ,let set π 0 =1 as installed by : Let A=A 0 + A 1 + A 2 and π A A=0. The following condition ought to be held for a QBD process: 4. The successive components of the solution can be easily generated to build stationary probability vector by using Matlab Packages as illustrated in ? .

System Characteristics and mathematical calculations:
The stability condition for the proposed system ? can be given by: where and Where E[S]: mean service time, and E [A]: mean inter arrival time, the system is stable when traffic intensity ρ < 1, it means that ratio average arrival ρ= λ mean Cµ mean , (ρ must be < 1).

Performance measurements:
• The probability that there are k packets existing within the queuing system can be expressed as: it can be argued that the probability of the system is not busy when it is at: • The average number of packets in Ph/Ph/C queuing system is given by: • The packets average number waiting within the queue may be given by: • The average response time is expressed as: • The average time of arrival packets waiting in system may be expressed as: • Mean queue length is defined as : where : L = L q + L S and L q = ρL and L = E[V ], L denotes the total length of the queue , L q is the queue arrival length, L S denotes the queue service length, and L s = ∑ ∞ k=1 KP k .
• • The throughput of the queuing system is defined as the departure rate X=λ if all arrived packets get their service and leave the SD-WAN controller system 16 . The properties of the queuing system are given in Table 1.

Markov process and state transitions diagram
The process can be visualized as a graph with the transition probabilities as illustrated in Fig 5 and Fig 6. • The describe of state in network (k, a, s) where: k denotes the number of packets in system, a denotes arrival phases of arriving packet and s denotes current phases of service 17 .

Simulation and results:
Through the present paper, efficiency, effectiveness as well as stability of the proposed SD-WAN network topology are considered. In this section the performance measurement is studied by MATLAB simulate and the results are persecuted and discussed. It was resorted to studying the SD-WAN network to improve the performance of wide area networks and reduce lost packages, as more than one branch is connected to the main canter. There are many factors affecting network performance and the speed of data transmission between WAN branches such as: utilization factor (system load), arrival rate, system size, system time Queue size and Queue time. In the following, several results obtained from the previous study will be presented. The Fig 7  refers to the improvement of SD-WAN throughput compared to traditional networks, as the less the system utilization, the higher the throughput.
The second result as shown Fig 8 the delay in traditional networks is much greater than SD-WAN networks, as in the experiment of this paper the delay in SD-WAN networks is less than (1.2 ms) than traditional networks , and therefore the system response speed is less in traditional networks than SD-WAN networks.
Also, one of the most important results that have been reached is as shown in Fig 9 that the queuing time in traditional networks is greater than in SD-WAN.  The number of arrival packet in traditional networks is large, and therefore the system utilization is high. Accordingly, the waiting packets will be large in traditional networks compared to the SD-WAN as Fig 10. The Fig 11 shows that with the decrease in the utilization system, the service rate increases, and thus, the service time will be reduced with reduced arrival rate.
At the end the service rate in SD-WAN improved by a very significantly due to the separation of the control plan from the data plan, because to the number of operations decreased and the consumption of network resources as shown in Fig 12.

Conclusion
The SD-WAN network has some limitations that need to be taken into consideration, such as congestion in arrival packets, and this will lead to a delay in entering the packet to take the service, and thus a Queuing delay is the time a arrival packet waits in a queue until it can be serviced will occur in the network. This paper is motivated to propose an objective for modelling together with analysis systems for PH/PH/C queues aids in minimizing their problems, testing efficiency of the proposed SD-WAN