排队论的matlab仿真(包括仿真代码).docx

1、Wireless NetworkExperiment Three:Queuing TheoryABSTRACTThis experiment is designed to learn the fundamentals of the queuing theory. Mainly about the M/M/S and M/M/n/n queuing models.KEY WORDS: queuing theory, M/M/s, M/M/n/n, Erlang B, Erlang C.INTRODUCTIONA queue is a waiting line and queueing theor

2、y is the mathematical theory of waiting lines. More generally, queueing theory is concerned with the mathematical modeling and analysis of systems that provide service to random demands. In communication networks, queues are encountered everywhere. For example, the incoming data packets are randomly

3、 arrived and buffered, waiting for the router to deliver. Such situation is considered as a queue. A queueing model is an abstract description of such a system. Typically, a queueing model represents (1) the systems physical configuration, by specifying the number and arrangement of the servers, and

4、 (2) the stochastic nature of the demands, by specifying the variability in the arrival process and in the service process. The essence of queueing theory is that it takes into account the randomness of the arrival process and the randomness of the service process. The most common assumption about t

5、he arrival process is that the customer arrivals follow a Poisson process, where the times between arrivals are exponentially distributed. The probability of the exponential distribution function is ft=e-t.l Erlang B modelOne of the most important queueing models is the Erlang B model (i.e., M/M/n/n

6、). It assumes that the arrivals follow a Poisson process and have a finite n servers. In Erlang B model, it assumes that the arrival customers are blocked and cleared when all the servers are busy. The blocked probability of a Erlang B model is given by the famous Erlang B formula,where n is the num

7、ber of servers and A=/ is the offered load in Erlangs, is the arrival rate and 1/ is the average service time. Formula (1.1) is hard to calculate directly from its right side when n and A are large. However, it is easy to calculate it using the following iterative scheme:l Erlang C modelThe Erlang d

8、elay model (M/M/n) is similar to Erlang B model, except that now it assumes that the arrival customers are waiting in a queue for a server to become available without considering the length of the queue. The probability of blocking (all the servers are busy) is given by the Erlang C formula,Where =1

9、 if An and =An if An. The quantity indicates the server utilization. The Erlang C formula (1.3) can be easily calculated by the following iterative schemewhere PB(n,A) is defined in Eq.(1.1).DESCRIPTION OF THE EXPERIMENTS1. Using the formula (1.2), calculate the blocking probability of the Erlang B

10、model. Draw the relationship of the blocking probability PB(n,A) and offered traffic A with n = 1,2, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Compare it with the table in the text book (P.281, table 10.3).From the introduction, we know that when the n and A are large, it is easy to calculate the blo

11、cking probability using the formula 1.2 as follows. PBn,A= APB(n-1,A)m+APB(n-1,A)it use the theory of recursion for the calculation. But the denominator and the numerator of the formula both need to recurs( PBn-1,A) when doing the matlab calculation, it waste time and reduce the matlab calculation e

12、fficient. So we change the formula to be : PBn,A= APB(n-1,A)n+APB(n-1,A)=1n+APBn-1,AAPBn-1,A=1(1+nAPBn-1,A) Then the calculation only need recurs once time and is more efficient.The matlab code for the formula is: erlang_b.m%*% File: erlanb_b.m % A = offered traffic in Erlangs. % n = number of trunc

13、ked channels. % Pb is the result blocking probability. %*function Pb = erlang_b( A,n ) if n=0 Pb=1; % P(0,A)=1 else Pb=1/(1+n/(A*erlang_b(A,n-1); % use recursion erlang(A,n-1) endendAs we can see from the table on the text books, it uses the logarithm coordinate, so we also use the logarithm coordin

14、ate to plot the result. We divide the number of servers(n) into three parts, for each part we can define a interval of the traffic intensity(A) based on the figure on the text books : 1. when 0n10, 0.1A10.2. when 10n20, 3A20.3. when 30n100, 13A120.For each part, use the “erlang_b” function to calcul

15、ate and then use “loglog” function to figure the logarithm coordinate.The matlab code is :%*% for the three parts.% n is the number servers.% A is the traffic indensity.% P is the blocking probability.%*n_1 = 1:2;A_1 = linspace(0.1,10,50); % 50 points between 0.1 and 10.n_2 = 10:10:20;A_2 = linspace

16、(3,20,50); n_3 = 30:10:100; A_3 = linspace(13,120,50); %*% for each part, call the erlang_b() function.%*for i = 1:length(n_1) for j = 1:length(A_1) p_1(j,i) = erlang_b(A_1(j),n_1(i); end end for i = 1:length(n_2) for j = 1:length(A_2) p_2(j,i) = erlang_b(A_2(j),n_2(i); end end for i = 1:length(n_3)

17、 for j = 1:length(A_3) p_3(j,i) = erlang_b(A_3(j),n_3(i); end end %*% use loglog to figure the result within logarithm coordinate.%*loglog(A_1,p_1,k-,A_2,p_2,k-,A_3,p_3,k-);xlabel(Traffic indensity in Erlangs (A) ylabel(Probability of Blocking (P) axis(0.1 120 0.001 0.1) text(.115, .115,n=1) text(.6

18、, .115,n=2) text(7, .115,10) text(17, .115,20) text(27, .115,30) text(45, .115,50) text(100, .115,100) The figure on the text books is as follow:We can see from the two pictures that, they are exactly the same with each other except that the result of the experiment have not considered the situation

19、 with n=3,4,5,12,14,16,18.2. Using the formula (1.4), calculate the blocking probability of the Erlang C model. Draw the relationship of the blocking probability PC(n,A) and offered traffic A with n = 1,2, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. From the introduction, we know that the formula 1.4 i

20、s : PCn,A=nPB(n,A)n-A(1-PB(n,A)Since each time we calculate the PBn,A, we need to recurs n times, so the formula is not efficient. We change it to be: PCn,A=nPB(n,A)n-A(1-PB(n,A)=1n-A(1-PB(n,A)nPB(n,A)=1(An+n-AnPBn,A) Then we only need recurs once. PBn,A is calculated by the “erlang_b” function as s

21、tep 1.The matlab code for the formula is : erlang_c.m%*% File: erlanb_b.m % A = offered traffic in Erlangs. % n = number of truncked channels. % Pb is the result blocking probability. % erlang_b(A,n) is the function of step 1.%*function Pc = erlang_c( A,n ) Pc=1/(A/n)+(n-A)/(n*erlang_b(A,n);endThen

22、to figure out the table in the logarithm coordinate as what shown in the step 1.The matlab code is :%*% for the three parts.% n is the number servers.% A is the traffic indensity.% P_c is the blocking probability of erlangC model.%*n_1 = 1:2;A_1 = linspace(0.1,10,50); % 50 points between 0.1 and 10.

23、n_2 = 10:10:20;A_2 = linspace(3,20,50); n_3 = 30:10:100; A_3 = linspace(13,120,50); %*% for each part, call the erlang_c() function.%*for i = 1:length(n_1) for j = 1:length(A_1) p_1_c(j,i) = erlang_c(A_1(j),n_1(i); %erlang_c end end for i = 1:length(n_2) for j = 1:length(A_2) p_2_c(j,i) = erlang_c(A

24、_2(j),n_2(i); end end for i = 1:length(n_3) for j = 1:length(A_3) p_3_c(j,i) = erlang_c(A_3(j),n_3(i); end end %*% use loglog to figure the result within logarithm coordinate.%*loglog(A_1,p_1_c,g*-,A_2,p_2_c,g*-,A_3,p_3_c,g*-);xlabel(Traffic indensity in Erlangs (A) ylabel(Probability of Blocking (P

25、) axis(0.1 120 0.001 0.1) text(.115, .115,n=1) text(.6, .115,n=2) text(6, .115,10) text(14, .115,20) text(20, .115,30) text(30, .115,40)text(39, .115,50) text(47, .115,60)text(55, .115,70)text(65, .115,80)text(75, .115,90)text(85, .115,100)The result of blocking probability table of erlang C model.T

26、hen we put the table of erlang B and erlang C in the one figure, to compare their characteristic. 10010-1The line with * is the erlang C model, the line without * is the erlang B model. We can see from the picture that, for a constant traffic intensity (A), the erlang C model has a higher blocking p

27、robability than erlang B model. The blocking probability is increasing with traffic intensity. The system performs better when has a larger n.ADDITIONAL BONUSWrite a program to simulate a M/M/k queue system with input parameters of lamda, mu, k.In this part, we will firstly simulate the M/M/k queue

28、system use matlab to get the figure of the performance of the system such as the leave time of each customer and the queue length of the system.About the simulation, we firstly calculate the arrive time and the leave time for each customer. Then analysis out the queue length and the wait time for each customer use “for” loops. Then we let the input to be lamda = 3, mu = 1