M. F. Driscoll and N. A. Weiss discussed the modeling and solution of problems concerning motel reservation networks in “An Application of Queuing The

Jason Farmer 2021-03-07 Answered
M. F. Driscoll and N. A. Weiss discussed the modeling and solution of problems concerning motel reservation networks in “An Application of Queuing Theory to Reservation Networks” (TIMS, Vol. 22, No. 5, pp. 540–546). They defined a Type 1 call to be a call from a motel’s computer terminal to the national reservation center. For a certain motel, the number, X, of Type 1 calls per hour has a Poisson distribution with parameter λ=1.7.
Determine the probability that the number of Type 1 calls made from this motel during a period of 1 hour will be:
a) exactly one.
b) at most two.
c) at least two.
(Hint: Use the complementation rule.)
d. Find and interpret the mean of the random variable X.
e. Determine the standard deviation of X.
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Expert Answer

Velsenw
Answered 2021-03-08 Author has 91 answers

Step 1
Given:
λ=1.7
Formula Poisson probability:
P(X=k)=λkeλk!
Complement rule:
P(¬ A)=1  P(A)
Addition rule for mutually exclusive events (special addition rule):
P(A or B)=P(A) + P(B)
Step 2
SOLUTION
a) Evaluate the formula of Poisson probability at k=1:
P(X=1)=1.71e1.71! 0.3106
Step 3
b) Evaluate the formula of Poisson probability at k=0, 1, 2:
P(X=0)=1.70e1.70! 0.1827
P(X=1)=1.71e1.71! 0.3106
P(X=2)=1.72e1.72! 0.2640
Use the special addition rule:
P(X2)=P(X=0)+P(X=1)+P(X=2)
=0.1827+0.3106+0.2640
=0.7572
Step 4
c) Evaluate the formula of Poisson probability at k=0,1,2:
P(X=0)=1.70e1.70! 0.1827
P(X=1)=1.71e1.71! 0.3106
Use the special addition rule:
P(X<2)=P(X=0)+P(X=1)
=0.1827+0.3106
=0.4932
Use the complement rule:
P(X2)=1P(X<2)=10.4932=0.5068
Step 5
d) The meanof the Poisson distribition is equal to the value of the parameter λ
μX=λ=1.7
On average, there are 1.7 Type 1 calls are made from this motel during a period of 1 hour.
Step 6 e) The variance of the Poisson distribution is equal to the value of the parameter λ
σX2=λ=1.7
The standard deviation is the square root of the variance:
σX=σX2=1.7 1.3038

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