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Queueing Theory 2. Nikolaos Limnios
Читать онлайн.Название Queueing Theory 2
Год выпуска 0
isbn 9781119755227
Автор произведения Nikolaos Limnios
Жанр Математика
Издательство John Wiley & Sons Limited
and
CONDITION 1.7.– Service times have the first exponential phase, i.e.
where
and independent random variables andAs regeneration points for Y we take subsequence
of the sequence such that at time interrupted services for processes are in the exponential phase. Because of conditions 1.6 and 1.7, Y is a strongly regenerative flow and we may define the common sequence of regeneration points for both processes X and Y with the help of formula [1.4]. We need only to take instead ofBecause of [1.10] we can easily obtain from the renewal theory the formula for the rate of the auxiliary process
Now we may calculate the traffic rate ρ and under some assumptions we get the necessary and sufficient stability condition for the system based on theorems 1.1 and 1.2. As an example, we consider the famous case (Morozov et al. 2011) when
i.e. a server may be in an available or unavailable state. Let be moments of breakdowns and moments of restorations for the ith server. HereThen
denote the length of the nth blocked and the nth available period for the ith server, respectively, The sequence consists of iid random vectors (for all ) and these sequences do not depend on the input flow X and service times. Let be the length of the nth cycle for the server i. A cycle consists of a blocked period followed by an available period. We assume thatWe put ni(t) = 0 if the ith server is in an unavailable state at time t and ni(t) = 1, otherwise
If a blocked period has an exponential phase, i.e. where are independent random variables and has an exponential distribution with a parameter αi, then we may define the sequence of regeneration points for the regenerative process as above. Therefore, condition 1.6 holds. Under condition 1.7, the auxiliary process Y is strongly regenerative and we can construct the common points of regeneration for X and Y and apply theorems 1.1 and 1.2 for this model. Sincewe have from [1.11]
If bi = b, then we get the same stability condition as obtained in Morozov et al. (2011) for a queueing system GI|G|m with a common distribution function of service times for all servers.
COROLLARY 1.1.– For a queueing system with
if ρ > I.
Under condition 1.4, the process is stochastically bounded if ρ < 1.
PROOF.– Let, as before,
be the number of customers actually served on the ith server up to time t. It is evident that stochastic inequalityfor t > 0 takes place and hence
Since
To prove the second statement, we first assume that conditions 1.6 and 1.7 hold. Then condition 1.1 for the process Y takes place. We also may organize the performance of the systems S and S0 in such a way that inequality [1.8] is realized when
Thus, conditions 1.1, 1.4 and 1.5 are satisfied and because of theorem 1.2 the process Q is stochastically bounded.If conditions 1.6 and 1.7 (or one of them) are not valid, we construct a system Sδ satisfying conditions 1.6 and 1.7 and majorising our system S, so that in distribution