“ n should be equal to or great than k” since if n is less than k (n<k), Pk(t)=0
Let’s think some customer C,
Let’s find P{C arrived at time x and in service at time t | x=(0,t)] } P{C arrives in (x, x+dx] | C arrives in (0, t] }P{C is in service | C arrives at x, and x = (0,t] }
Since theorem of Poisson Process,
The theorem is that Given that N(t) =n, the n arrival times S1, S2, …Sn have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on the interval (0, t)
Thus, P{C is in service | C arrives between time (0, t] } Since let y=t-x, x=0 → y=t, x=t →y=o, dy=-dx
Therefore,
Page 147 in “Fundamentals of Queuing Theory –Third Edition- , Donald Gross Carl M. Harris
a.
b.
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3.
a. let X=service time (Random variable) and XT=total service time (Random variable) X2=X+X, X3=X+X+X, ….. f2(x2) : pdf of X2, f3(x3) : pdf of X3, …… B2(x2) : Cdf of X2, B3(x3) : Cdf of X3, …… Let B1(x)=B(x)
b.
Solution of 3-b
c.
d.
Notation
Proof
b.
Or Let F,G, and H be non-decreasing function with non-negative domain and range. Also for scalar a and b
Since
b.
3. Let
+ +
The actual probabilities for priority-1 customer : Pn1
