Conclusion and Perspectives
It is now apparent that the renewal approach to fluid queues, based on the analogy with QBD processes, allows for the unified and straightfor- ward analysis of models which exhibit a variety of features.
Very interesting questions arise from the results presented in this work. The first one is to study feedback fluid queues with thresholds, as those presented in Sections 4.6 and 4.7, but allowing the net input rates to take any real value. The development is quite straightforward regarding the model of Section 4.6, but somewhat more involved for the model of Section 4.7.
In Chapter 5, we constructed a fluid queue with stationary indepen- dence between the level and the phase; we obtained this at the cost of removing the probability mass at level zero. One can construct other fluid queues which may exhibit the same independence property, for example by adding some probability mass on the states (0,S+). This construction requires a major modification of fluid queues.
We have exploited in [6] and [48] the relationship between risk pro- cesses and fluid queues, and obtained efficient algorithms for computing the probability of ruin under different scenarios. The method developed there also applies to more general jump processes: we have shown in Dzial et al. [19] how to associate a fluid queue with a jump process and derive some interesting results about the latter, at least in cases where transitions are level-independent. It would be interesting to investigate more complex situations.
A line of enquiring which has recently opened and which we have
166 Conclusion and Perspectives
not mentioned earlier is the so-called transient analysis of fluid queues, whereby one analyses quantities such as first passage time distributions.
Finally, even the analogy to QBD processes is far from being thor- oughly exploited; one might for instance analyse fluid queues with a pos- itive drift and determine if there exists an invariant measure for the dif- ferential equations, following a similar development for QBD processes.
Notations
X(·) : level of a fluid queue
X(b)(·) : level of a fluid queue with finite capacityb {ϕ(t) :t∈R+}: Markovian phase process
T : infinitesimal transition generator of{ϕ(t)}
ξ : steady state probability vector corresponding to the generator T S : state space of{ϕ(t)}
ri: net input rate of fluid when the phase is i r= (ri:i∈ S)
µ=ξr : mean stationary drift S0 ={i∈ S :ri= 0}
S+ ={i∈ S :ri>0}
S−={i∈ S :ri <0}
S• =S+∪ S−
s0,s+,s− : cardinalities of S0,S+ and S−, respectively Fi(x;t): joint distribution function of state (x, i) at timet fi(x;t) : joint density function of state(x, i) at timet πi(x) = limt→∞fi(x;t) : stationary density of state (x, i) π(x) = (πi(x) :i∈ S)
µ(x) =π(x)1: stationary density of level x
p,p(0) : steady state probability mass vector of the empty buffer p(b) : steady state probability mass vector of the full buffer
168 Notations
eK x : expected number of crossings of(x,S+), starting from(0,S+), before returning to level zero
Ψ: matrix of first return probabilities to the initial level
Ψ(+b−) : first return probabilities to the initial level, before reaching levelb
Λ(++b) : first passage probabilities to level b, starting from level zero, before returning to the initial level
N+(b)(0, x) : expected number of visits to levelx, starting from(0,S+), before visiting levels 0 andb
N−(b)(b, x) : expected number of visits to levelx, starting from(b,S−), before visiting levels 0 andb
{D(t)}: Markov process of downward records U : infinitesimal transition generator of {D(t)}
K,ˆ Ψ,ˆ Uˆ : equivalent to K,Ψ,U for the level-reversed fluid queue Ψˆ(−b+) : equivalent to Ψ(+b−) for the level-reversed fluid queue
Λˆ(−−b) : equivalent to Λ(++b) for the level-reversed fluid queue
A0,A1,A2 : transition matrices of a QBD process
G: first passage probabilities to lower levels for a QBD process R : expected number of visits to level one, starting from level zero,
before returning to level zero, for a QBD process G,ˆ Rˆ : equivalent to G,R for the level-reversed QBD
sp(M): spectral radius of a matrix M M#: group inverse of a matrix M R(s) : real part of a complex numbers 0 : vector of zeros
1 : vector of ones I : identity matrix
