Queueing theory

The current research in cognitive radio has been considering absolute guarantee for primary users allowing secondary users access to spectrum only if there is no primary users with data to send. At high arrival rate of primary users this might lead to complete starvation of secondary users and yet it is possible to release some spectrum to secondary users by delaying primary users without affecting their quality. We propose a resource allocation scheme that uses delayed time periods of primary users to transmit secondary user's data packets without jeopardizing the quality of primary users. We analytically modeled the scheme using M/G/1 queue. Our numerical experiments demonstrate that secondary users can be offered some quality of service by delaying primary users in the system to a limit that does not degrade their performance.

We propose a novel explicit rate-flow-control algorithm intended for available-bit-rate (ABR) service on an ATM network subject to loss and fairness constraints. The goal is to guarantee low cell loss in order to avoid throughput collapse due toretransmission by higher level protocols. The mechanism draws on measuring the current queue length and bandwidth availability, as well as tracking tile current number of active sessions contending for capacity, to adjust an explicit bound on the source transmission rates. We identify the factors that affect queue overflows and propose simple design rules aimed at achieving transmission with controlled loss in a dynamic environment. We also discuss how conservative design rules might be relaxed by accounting for statistical multiplexing in bandwidth sharing among bursty ABR sources and variable-bit-rate (VBR) sources. Index Terms-ABR service, ATM networks, delay differential equations, explicit rate flow control. |. INTRODUCTION A SYNCHRONOUS transfer mode (ATM) networks are geared at supporting and integrating a variety of communication services. These services may be divided into those based on reservation, e.g., constant-and variable-bit-rate (CBR, VBR) services, and best effort services, such as unspecified-bit-rate (UBR) and to some extent available-bit-rate (ABR) services. Among the latter, ABR service promises to play an important role in supporting high-bandwidth data as well as Internet traffic, such as TCP traffic. The rationale for including ABR is to provide an economical and flexible way to carry data traffic, as might be needed to simplify adoption of ATM to support delay adaptive real-time applications [6]. From the service provider's point of view, ABR traffic can be used to enhance utilization by directing sources to make the most of the network's available capacity subject to minimum cell rate and cell loss guarantees. It has been shown in [22], [23] that TCP performs poorly over ATM networks when congestion leads to lost ATM cells which in turn result in multiple corrupted IP packets. Significant performance degradation results from corrupted IP packets since

This paper examines the relationship between lot release variability and queuing time in a simple six operation manufacturing process line operating under a push-type lot release policy. Five different lot release strategies are modelled, based on two different operation capacity scenarios. The performance resulting from the departure variability caused by each of the release strategies is analyzed, as well as the queuing impact due to capacity.

Efficient implementations of Dijkstra's shortest path algorithm are investigated. A new data structure, called the radix heap , is proposed for use in this algorithm. On a network with n vertices, m edges, and nonnegative integer arc costs bounded by C , a one-level form of radix heap gives a time bound for Dijkstra's algorithm of O ( m + n log C ). A two-level form of radix heap gives a bound of O ( m + n log C /log log C ). A combination of a radix heap and a previously known data structure called a Fibonacci heap gives a bound of O ( m + n a @@@@log C ). The best previously known bounds are O ( m + n log n ) using Fibonacci heaps alone and O ( m log log C ) using the priority queue structure of Van Emde Boas et al. [ 17].

Crossbar architectures are widely used to implement highperformance network switches and routers. A crossbar switch can transfer cells between multiple ports simultaneously by closing multiple cross points. This crossbar configuration must be determined by an intelligent centralized scheduler that can ensure fairness and high utilization. In this paper, we describe the design and implementation of two symmetric scheduling algorithms for configuring crossbars in input-queued switches that support virtual output queueing. Our target is a fast packet switch that can support 32 ports, each operating at 20Gbps. Using a 0.35m CMOS process, the faster of the two schedulers is capable of configuring a 3232 crossbar once every 12.61ns. Our scheduler designs are based on a two-dimensional ripple carry arbiter architecture. To ensure fairness, both architectures support a round-robin priority rotation scheme. Based on network simulations, we show that our arbiter designs can achieve near optimal system performance.

The input-queued switch architecture is widely used in Internet routers due to its ability to run at very high line speeds. A central problem in designing an input-queued switch is the scheduling algorithm that decides which packets to transfer from ingress ports to egress ports in a given timeslot. It is desirable that such algorithms be iterative (so as to be pipelineable), distributed (allowing flexibility in hardware implementation) and are able to deliver high performance (in terms of throughput and delay). In practice, implementable algorithms have so far had limited success in combining all of the above properties. For example, the popular iSLIP [1] algorithm is known to perform suboptimally, but it is commercially deployed mainly because it is iterative and distributed. The main contribution of this paper is the design and systematic analysis of two algorithms which, to the best of our knowledge, are the first high-performance iterative and distributed scheduling algorithms with possibility of efficient implementation. We first present an iterative, distributed and low-delay maximal throughput algorithm based on the celebrated "Auction Algorithm" [2], [3]. This algorithm can be seen as a natural extension of iSLIP when queue-size information is allowed to be exchanged. The standard auction algorithm can take an unbounded number of iterations to converge in the worst case. However we show that under admissible Bernoulli i.i.d. traffic, our algorithm takes O(n 2) iterations, where n is the number of ingress/egress ports in the switch. Moreover for a switch with finite buffer-size, the algorithm allows for a graceful trade-off between running time and performance, which we verify by representative simulation results. Next, we propose and analyze a throughput-optimal, iterative and distributed scheduling algorithm influenced by Max-Product Belief Propagation [4], [5]. Recently the problem of efficient transmission over multi-hop wireless networks has been formulated as that of finding an appropriate schedule over the gridgraph abstraction of the network. A key feature of the multi-hop wireless transmission problem is that while the communication subgraph is bipartite, the bi-partition is allowed to change in each scheduling epoch. We show that our algorithm can be used to efficiently schedule traffic in multi-hop wireless networks.

Large-scale network simulation is widely used to facilitate development, testing and validation of new and existing network technologies. To ensure a high-fidelity experimental environment, we often need to embed real devices and have the simulator running faster than real time. Since the generation and movement of background traffic in a network simulation represents so much of the workload, we develop here techniques for modeling background traffic through switches that use Fair Queueing scheduling. Our work is an extension of earlier efforts that assumed all switches use First-Come-First-Serve scheduling. It turns out the the scheduling policy has an important impact on the logic of the earlier technique, and on the performance it delivers. We describe the algorithm and give experimental results that show that like the earlier work, very significant acceleration of background traffic simulation is achieved.

The present survey is based on the principle of retrials in queueing theory i.e. the customers who repeatedly try to avail the services from the server, on finding dejection, because of the inactivity of the server to provide service at an instant of selection from the queue. A bunch of work has previously been done on this front from past since it is realistic phenomena of many service system. The current investigation aims to present a brief review of the major works done on retrial queueing systems in recent years. The bibliography consists of research articles which were published in journals of repute during the period 2010-2015. In addition to the above references, an exhaustive list of books and survey papers on retrial queueing systems is also prepared so as to provide a detailed enough catalog for further understanding and research in retrial queueing domain. We have classified the journal papers according to the year of publishing. The aim of the present study is to turn up at a broad enough for a collection of important results in the theory of retrial queueing systems and their applications in solving several realistic problems. The main contribution is the bibliography of recent papers and books collected from different databases and private sources. At last, we have arranged some of the references according to the classification based on analysis and solution technique. The author(s) hope that this survey paper could be of help to learners contemplating research on retrial queues.

Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail probabilities are not necessarily exponential (as in most 'traditional' queueing systems). Further, we show the impact and significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results with simulations.

In this paper, we consider a multicommodity flow problem where for each pair of vertices, (u,v), we are required to sendf half-units of commodity (uv) from u to v and f half-units of commodity (vu) from v to u without violating capacity constraints. Our main result is an algorithm for performing th9 task provided that the capacity of each cut exceeds the demand across the cut by a b(log n) factor. The condition on cuts is required in the worst case, and is trivially within a i(log n) factor of optimal for any flow problem.

We present a model to describe the inbound air traffic over a congested hub and we show that this model gives a very accurate description of the traffic by comparing our theoretical distribution of the queue with the actual distribution observed at Heathrow airport. We also discuss the robustness of our model.

This book, written in Portuguese, deals with Operational Research topics, such as linear programming models and direct techniques for solving Ax = b systems, revised simplex method, Markov processes and queuing systems, in addition to Monte Carlo simulation.

Traffic models play an important role in both today's traffic research and in many traffic applications such as traffic flow prediction, incident detection and traffic control. Modelling traffic dynamics and optimising the control signal are two interrelated problems. Modelling provides fundamental understanding of traffic dynamics and behaviour. In this paper, traffic signal is modelled as a M/M/l queueing theory. The validation of a simulation model (M/M/l queue) with different arrival rates is presented. From the result, a traffic light model was developed by applying M/M/l queue theory for single intersection. In the optimisation strategy, the Markov decision control is applied to minimize queue length and waiting time. Simulation results show the excellent potential of this approach

With the current shift toward a service prominent economy, there exists a necessity in continuous improvements of service provision. Hospitality services have emerged as a foremost service under these circumstances. A university cafeteria is among the most time intensive services within a university premise. The tight schedules of the students and the staff influence the rushed atmosphere at the cafeterias that leads to customer dissatisfaction and brand switching. The study identified the congession at a hostel cafeteria of a state university. The objective of the study was to improve performance by reducing in the waiting times. The cafeteria was studied and primary data were collected during the rushed hours of weekdays for two weeks. The obtained data were simulated with Rockwell ARENA 14.5 version. Existing system allowed a waiting time of 3.30 minutes in the queue. The total time in the system was 4.81 minutes. Two model developments of increasing the service provision and the addition of another counter were simulated on the software. The waiting times resulted in the simulations depicted the efficiency in the increment of the service provision by twofold at the counter. The simulations for the data available depicted the addition of resources to the counter as the most efficient alternative. It obtained a reduction in waiting time by 74.84 percent and the service provision was increased by 67.54 percent in the system. The developments can be acheived by the standardization of the food serving and the payment handling at the counters.

The IEEE 802.11ai is an upcoming fast initial link setup (FILS) amendment that could enable a STA to achieve secure link setup in less than 100 ms. A successful link setup process will then allow the STA to send IP traffic with a valid IP address through the AP. In this paper, we first present a performance analysis on how well the legacy 802.11 can support FILS. We then demonstrate that the rate at which the medium saturates is mainly dependent on the offered load which has strong dependencies on the active scan rates (i.e., amount of transmitted probe request frames) and number of responding APs (i.e., amount of transmitted probe response frames) among others. Moreover, the average active scanning duration per channel could be maintained below 5 ms provided the medium is not saturated. Our results also indicate that a combination of the enhanced distributed channel access (EDCA) procedure and the proposed active scanning enhancements provides at least 20% and up to 250% improvements in supporting the functional requirements of FILS as compared to the 802.11 EDCA and legacy distributed coordination function (DCF), respectively.

- ATM enables the customers perform financial transactions without the need for a human cashier, clerk or bank teller. Queuing is the activity where customers avail the desired service, which could be processed or distributed one at a time. Thus, to maintain the customers, the service time needs to be improved. In this paper we analyze how queuing theory can be used to accurately predict the arrival rate of the customers and service rate of service facility. We collected data from an ATM in the city. The M/M/1 queuing model has been used to obtain the average arrival rate and the average service rate . The results of the analytic model are compared with the simulation model developed using MATLAB. The benefits of a queuing model are also discussed.

3. CODINGAN :
#include <stdio.h>
#include <conio.h>

typedef struct
{
            int data[100];
            char data1[100][20];
            char data2[100][30];
            char data3[100][15];
            int depan;
            int belakang;
}Queue;
Queue antrian;
Queue no;
Queue nama;
Queue poli;

int pilihan, data, i, j;
char data1[20], data2[30], data3[15];

int isEmpty()
{
            if (antrian.belakang==-1)
                        return 1;
            else
                        return 0;
}

int isFull()
{
            if(antrian.belakang==100)
                     

return 1;
            else
                        return 0;
}

void Enqueue(int data, char data1[20], char data2[30], char data3[15])
{
            if(isEmpty()==1)
            {
                        antrian.depan=antrian.belakang=0;
                        nama.depan=nama.belakang=0;
                        poli.depan=poli.belakang=0;
                        no.depan=no.belakang=0;
                        antrian.data[antrian.belakang]=data;
                        for(i=0;i<20;i++)
                        {nama.data1[nama.belakang][i]=data1[i];}
                        for(i=0;i<30;i++)
                        {poli.data2[poli.belakang][i]=data2[i];}
                        for(i=0;i<15;i++)
                        {no.data3[no.belakang][i]=data3[i];}
                        printf("data anda telah masuk. silahkan menunggu panggilan");
            }
            else if(isFull()==0)
            {
                        antrian.belakang++;
                        nama.belakang++;
                        poli.belakang++;
                        no.belakang++;
                        antrian.data[antrian.belakang]=data;
                        for(i=0;i<20;i++)
                        {nama.data1[nama.belakang][i]=data1[i];}
                        for(i=0;i<30;i++)
                        {poli.data2[poli.belakang][i]=data2[i];}
                        for(i=0;i<15;i++)
                        {no.data3[no.belakang][i]=data3[i];}
                        printf("data anda telah masuk. silahkan menunggu panggilan");
            }
}

int Dequeue()
{
            if(isEmpty()==0)
            {
            int i,e ;
            char a[20], b[30];
         
  e=antrian.data[antrian.depan];
            for(i=0;i<20;i++)
            {
                        a[i]=nama.data1[nama.depan][i];
            }
            for(i=0;i<30;i++)
            {
                        b[i]=poli.data2[poli.depan][i];
            }
            for(i=antrian.depan;i<antrian.belakang;i++)
            {
                        antrian.data[i]=antrian.data[i+1];
                        for(j=0;j<20;j++)
                        {nama.data1[i][j]=nama.data1[i+1][j];}
                        for(j=0;j<30;j++)
                        {poli.data2[i][j]=poli.data2[i+1][j];}
                        for(j=0;j<15;j++)
                        {no.data3[i][j]=no.data3[i+1][j];}
            }
            antrian.belakang--;
            nama.belakang--;
            poli.belakang--;
            no.belakang--;
            printf("antrian no %i dengan nama %s silahkan memasuki poli %s !", e, a, b);
            return e;
            return a[20];
            return b[30];
            }
            else
            {
                        printf("antrian kosong");
            }
}

void buat()
{
            antrian.depan=antrian.belakang=-1;
            nama.depan=nama.belakang=-1;
            poli.depan=poli.belakang=-1;
            no.depan=no.belakang=-1;
}

int main()
{
            int n;
            buat();
         
  n=1;
            do
            {
                        ("cls");
                        puts("aplikasi antrian rumah sakit");
                        puts("============================\n");
                        puts(" menu :");
                        puts("1. masukkan antrian");
                        puts("2. panggil antrian");
                        puts("3. lihat antrian");
                        puts("4. exit");
                        printf("silahkan masukkan pilihan (1-4) : "); scanf("%i", &pilihan);
                        switch(pilihan)
                        {
                                    case 1 :
                                                printf("no antrian anda adalah %i\n", n);
                                                data=n;
                                                printf("masukkan no. rekam medis anda :"); scanf("%s",&data3);
                                                printf("masukkan nama lengkap anda(gunakan tanda (_) untuk spasi) :"); scanf("%s",&data1);
                                                printf("masukkan poli berobat anda (gunakan tanda (_) untuk spasi):"); scanf("%s",&data2);
                                                Enqueue(data, data1, data2, data3);
                                                n++;
                                    break;

                                    case 2 :
                                                Dequeue();
                                    break;

                                    case 3 :
                                                if(isEmpty()==1)
                                                {
                                                            printf("antrian kosong !");
                                                            break;
                                                }
                                                puts("no antrian          no.reka medis              nama lengkap              poli berobat");
                                                for(i=antrian.depan;i<=antrian.belakang;i++)
                                                {
                                                            printf("%i                    %s                  %s                  %s\n", antrian.data[i], no.data3[i], nama.data1[i], poli.data2[i]);
                                                }

                                    break;
                                 
  case 4 :
                                                printf("terimakasih atas partisipasinya !!!");
                                    break;
                                    default :
                                                puts("salah tuh pilihannya !!!");
                                    break;
                        }
                        getch();
            }while(pilihan!=4);

}

Queues are characterized structures fashioned to maintain order and create a hold on time, money and human contribution towards development and efficient performance of any system. Queues are experienced in our everyday activities. Queue causes inconvenience to individuals (patients) and economic costs to firms and organizations. Patients wait for minutes, hours, days or months to receive medical service-waiting before, during or after being served. Queuing theory is a mathematical approach to the study of waiting in lines/queues. This research presents the results of a study that evaluates the effectiveness of a queuing model in identifying the major bottleneck in healthcare operations. It uses chi-square and Erlang's queuing to analyse data collected from the University of Benin Health Center and the Faith Mediplex center over one month period. Results showed that queue characteristics existing at the healthcare centers during the situation analysis was a single server multiple queue model. However, after the study was done involving staff at the understudy healthcare centers, it was discovered that queuing is mainly found in record unit and doctor consultation waiting lobby. We finally came to a conclusion and made the some recommendations on how best healthcare centers can maximize the benefits of queuing model for good and effective operations.

This research intends to discuss effectiveness of queue management systems with regards to processing speed and convenience of applicants, its maximum potential and limits on how to improve the system itself and decrease delay time, and the impact of its application in operations and customer flow in offices.

Assigning and scheduling vehicle routes in a dynamic environment is a crucial management problem. Despite numerous publications dealing with efficient scheduling methods for vehicle routing, very few addressed the inherent stochastic and dynamic nature of travel times. In this paper, a vehicle routing problem with time-dependent travel times due to potential traffic congestion is considered. The approach developed introduces the traffic congestion component based on queueing theory. This is an innovative modelling scheme to capture the stochastic behavior of travel times as it generates an analytical expression for the expected travel times as well as for the variance of the travel times. Routing solutions that perform well in the face of the extra complications due to congestion are developed. These more realistic solutions have the potential to reduce real operating costs for a broad range of industries which daily face routing problems. A number of datasets are used to illustrate the appropriateness of the novel approach. Moreover it is shown that static (or time-independent) solutions are often infeasible within a congested traffic environment which is generally the case on European road networks. Finally, the effect of travel time variability (obtained via the queueing approach) is quantified for the different datasets.

This paper provides a method for approzimating the probability distributions of stationary statistics in FIFO single server queues. The method is based on the Wiener Hopf factorization technique, and is applied to semi-Markov queues where the underlying state space is of unlimited size. A particularly important case is a model for a statistical multiplezer where the net input process forms a stationary ergodic Gaussian discretetime stochastic process.

Queuing is an inevitable situation experienced in everyday life. In any service system, a queue forms whenever current demand exceeds the current capacity to serve. In this paper, an analysis of the queuing situation at the ticket windows in the Nigerian Railway Corporation Lagos Terminus was carried out. To achieve this, primary data was obtained by observation and an appropriate queuing model was used to analyze the system and calculated its performance measures. Results showed that increasing the number of servers lead to a decrease in the average waiting time of passengers in queue and a decrease in the average waiting time of passenger in the system. Therefore by applying Queuing theory, the waiting time of passengers can be determined and improved upon. The study therefore recommends that management in railway stations can use results from queuing analysis in making decisions that increase the efficiency of operations at their ticket windows.

is an analytical method of problem-solving and decision-making that is useful in the management of organizations. In operations research, problems are broken down into basic components and then solved in defined steps by mathematical analysis.

We study the 'Creation of Pooling in Inventory and Queueing Models'. This research consists of the study of sharing a scarce resource (such as inventory, server capacity, or production capacity) between multiple customer classes. This is called pooling, where the goal is to achieve cost or waiting time reductions. For the inventory and queueing models studied, both theoretical, scientific insights are generated, as well as strategies which are applicable in practice.

This monograph consists of two parts: pooling and polling . In the first part, pooling is applied to multi-location inventory models. It is studied how cost reduction can be achieved by the use of stock transfers between local warehouses, so-called lateral transshipments. In this way, stock is pooled between the warehouses. The setting is motivated by a spare parts inventory network, where critical components of technically advanced machines are kept on stock, to reduce down time durations. We create insights into the question when lateral transshipments lead to cost reductions, by studying several models.

Firstly, a system with two stock points is studied, for which we completely characterize the structure of the optimal policy, using dynamic programming. For this, we formulate the model as a Markov decision process. We also derived conditions under which simple, easy to implement, policies are always optimal, such as a hold back policy and a complete pooling policy. Furthermore, we identified the parameter settings under which cost savings can be achieved. Secondly, we characterize the optimal policy structure for a multi-location model where only one stock point issues lateral transshipments, a so-called quick response warehouse. Thirdly, we apply the insights generated to the general multi-location model with lateral transshipments. We propose the use of a hold back policy, and construct a new approximation algorithm for deriving the performance characteristics. It is based on the use of interrupted Poisson processes. The algorithm is shown to be very accurate, and can be used for the optimization of the hold back levels, the parameters of this class of policies. Also, we study related inventory models, where a single stock point servers multiple customers classes.

Furthermore, in the first part, the pooling of server capacity is studied. For a two queue model where the head-of-line processor sharing discipline is applied, we derive the optimal control policy for dividing the servers attention, as well as for accepting customers. Also, a server farm with an infinite number of servers is studied, where servers can be turned off after a service completion in order to save costs. We characterize the optimal policy for this model.

In the second part of the thesis, polling models are studied, which are queueing systems where multiple queues are served by a single server. An application is the production of multiple types of products on a single machine. In this way, the production capacity is pooled between the product types. For the classical polling model, we derive a closed-form approximation for the mean waiting time at each of the queues. The approximation is based on the interpolation of light and heavy traffic results. Also, we study a system with so-called smart customers, where the arrival rate at a queue depends on the position of the server. Finally, we invent two new service disciplines (the gated/exhaustive and the k-gated discipline) for polling models, designed to yield 'fairness and efficiency' in the mean waiting times. That is, they result in almost equal mean waiting times at each of the queues, without increasing the weighted sum of the mean waiting times too much.


Slides of presentation during PhD defense: http://tue.academia.edu/SandraVanWijk/Talks/82570/Pooling_and_Polling_Creation_of_Pooling_in_Inventory_and_Queueing_Models

Prediction using different machine learning approaches have been applied in the last few decades in different areas and research fields. Waiting time is an undeniable fact for every queue and it is very important to develop a system that predicts its duration in real life with minimum error. In this paper we applied several machine learning algorithms and among them we chose Support Vector Regression (SVR) in a real-life Banking queue dataset that contains real-life queues of multiple Banks where we predicted waiting time for everyone in the queue. Moreover, we have compared the result of prediction using SVR with different classifications and clustering methods such as K-nearest-neighbor and K means Clustering. We have shown the feasibility of applying SVR in prediction of waiting time in banking queues of developing countries for everyone, which is applicable, and it performs well in queue analysis.

In this paper we consider an unreliable single server retrial queue accepting two types of customers, with negative arrivals, preemptive resume priorities and vacations. A distinguishing feature of the model is that the rates of the Poisson arrival process depends on the server state. For this model we investigate the stability conditions and the joint queue length distribution in steady state. We also prove that our model satisfies the stochastic decomposition property. Transient, as well as steady state solutions for reliability measures are obtained. Finally, numerical results demonstrate the typical features of the model under consideration.

Waiting is inevitable in any service organization. Hence queues are formed. A long queue makes customer dissatisfied whereas increasing server to decrease the length of a queue requires costs. Waiting Line Theory provides a platform to that notion. The health sector of Bangladesh has been facing a bad experience for any citizen of Bangladesh. One of the reasons behind this is the queuing system prevailing in this sector. This paper presents an investigation into the prevailing queuing system in the private practices of Dhaka city. On the basis of investigation suggestions are put forward to improve the situation.

A queueing system with a single server providing two stages of service in succession is considered. Every customer receives service in the first stage and in the sequel he decides whether to proceed to the second phase of service or to depart and join a retrial box from where he repeats the demand for a special second stage service after a random amount of time and independently of the other customers in the retrial box. When the server becomes idle, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service times are arbitrarily distributed. For such a system the stability conditions and the system state probabilities are investigated both in a transient and in a steady state. A stochastic decomposition result is also presented. Numerical results are finally obtained and used to investigate system performance.