Page 187 - Contributed Paper Session (CPS) - Volume 3

CPS1988 Saggou Hafida H. et al.
vacation time and if () = 6 , we define () as the elapsed second
6
verification delay time.

3. Analysis of the steady-state probabilities
We investigate in this section the steady-state distribution of the system.
The conditional completion rates for the repeated attempts of transit
customers, for the service of transit customers, for the service of recurrent
customers, for the first verification delay, for repair, for vacation and for second
verification delay times are given.

Theorem 1. Under the stability condition, the marginal Probability Generating
Functions of the server’s state queue size distribution are as follow:

( + )[( + ) − ∅() 0 ] 1 − ( + )
() =
0 ( + )
(1 − ( + ))∅() − ( + )[ − ( + )()]

( + ) ( + )[(1 − ()) + (1 − 2 ())] 0 1 − 1 ()
() =
1
(1 − ( + ))∅() − ( + )[ − ( + )()] ()

( + ) ( + )[() − ] 0 −1 1 − 2 ()
() =
2
(1 − ( + ))∅() − ( + )[ − ( + )()] ()

( + ) ( + )[(1 − ()) + (1 − 2 ())] 0 1 − 1 () 1 − 1 (())
() = ×
1,1
(1 − ( + ))∅() − ( + )[ − ( + )()] () ()

( + ) ( + )[() − ] 0 −1 1 − 2 () 1 − 1 (())
1,2 () = () × ()
(1 − ( + ))∅() − ( + )[ − ( + )()]

2,1 ()

2,2 ()

1 ()

( + ) ( + )[(1 − ()) + (1 − 2 ())] 0 1 − (())
() = (1 − ) ()
V s 1
(1 − ( + ))∅() − ( + )[ − ( + )()] ()

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