Page 118 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes distribution, Poisson distribution and etc are quite common. Poisson distribution arrival
is provided as below:
In (t, t+s), the probability of entity number k is:
–λs
e (λs) k
P{N(t+s)–N(t)=k}=
k!
In the formula, N(t) is the number of entity arrival in (0,t). t >=0, s>=0, k=0, 1, 2,  is the
arrival velocity. If the entity arrival satisfies steady Poisson distribution, arrival interval
will obey Index distribution and density function is:


f(t)= e – t  1 e –1/ ,t  0 = 1


 
average value of arrival interval.
2. Service Mode: Its character is that its server may be single or multiple, and service time
distribution is nothing about time or something about time, and server’s service time is
certain or random. Random service time is described with probability distribution, for
instance Normal distribution,

1 (x– ) 2
,
f(t)= e – , –       0
2 2 2
In the equation above, t is the time of server for each custom, which is obeying Normal
2
distribution, and the average µ, the variance is  .
3. Queuing Rule and the Criteria of the Queuing System: There are some queuing rules such
as FCFS, Random served, priority served and SCFS, etc. With studying the performances
of the queuing system, some criteria usually used are as below:
(a) Steady-state means delaying time d:
n
d= lim  D i /n
n
i 1
In the equation, D means NO.i entity’s delaying time, i.e. waiting time in the queue; n is
i
the number of the accepted entities; d is the mean time of waiting time of the n entities; D
i
the staying time of the entity in the system w:
n n
w= lim  w i /n  lim  (D  S i )/n
i
n n
i 1 i 1
In the equation as above, W is the staying time of No.i entity in the system, and equals to
i
the sum of waiting time in the queue D and accepting service time S .
i i
(b) Steady-state means step-length Q:
T
Q= lim  Q(t)dt/T
T 0
In the formula, Q(t) is the length of the queue at t, and T is the simulate time of the
system.
(c) Steady-state entity mean number L:
T T
L=lim  L(t)dt/T  lim  Q(t)+S(t)dt/T
T  0 T  0
In the formula, L(t) is the number of the entity in the system at t, and equals to Q(t) and S(t).

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