4.2 Markovian Fluid Models
4.2.3 Busy period analysis of Markovian fluid models
In this section we briefly summarize the most essential results of [1] and [72] on the busy period analysis of fluid models, they will be necessary in Chapter7for the solution of priority queues.
As mentioned above, Ψis the phase transition probability matrix between the beginning and the end of the busy period. If the duration of the busy period is also of interest, we can introduce matrixΨ(t), the time dependent counterpart ofΨ. Entry(Ψ(t))i,j, i ∈ N+,j ∈ N−,t > 0is the joint probability that the duration of the busy period is less thantand the underlying Markov chain is in state jwhen the fluid level returns to0 given that it was in stateiwhen the busy period was initiated.
According to Theorem 1 of [72], theLSTofΨ(t), denoted byΨ∗(s)satisfies theNARE Ψ∗(s)Q−+Ψ∗(s) +Ψ∗(s)Q−−+Q++Ψ∗(s) +Q+− =2sΨ∗(s). (141)
68 skip-free processes
(Note that settings →0gives back (135)).
Let the random variableBdenote the length of the busy period of a canonical fluid queue characterized by matrixQgiven that the state probability vector of the backgroundCTMCis κ={κi,i=1, . . . ,N+}when the busy period starts.
Theorem 18. TheLSTof the busy period fB∗(s) =E(e−sB)is given by
fB∗(s) =κΨ∗(s)1. (142)
Proof. The theorem follows from the probabilistic interpretation ofΨ(t). Theorem 19. Thekth moment of the busy period is given by
E(Bk) =κ(−1)kΨ(k)1, (143)
whereΨ(0)=Ψand matricesΨ(k),k>0are defined recursively as (Q+++ΨQ−+)Ψ(k)+Ψ(k)(Q−−+Q−+Ψ)
=2kΨ(k−1)−
k−1 i
∑
=1k i
Ψ(i)Q−+Ψ(k−i). (144) Proof. (144) follows from routine derivations withΨ(k)= dk
dskΨ∗(s)|s=0.
Since (135) providingΨ(0)is aNAREand (144) providingΨ(k),k > 0is a Sylvester equa-tion, theLSTof the busy period and the moments can be obtained in a numerically efficient way. The distribution function in time domain, FB(t) = P(B < t) = κΨ(t)1is, however, more involved to calculate. One can rely on a generic numerical Laplace transform inversion procedure, but according to our experience they are not always reliable up to the machine precision, and need complex arithmetic. Instead, a simple and elegant procedure called Erlan-gizationis available [72], according to which the order-napproximationFB(n)(t)is
FB(n)(t) =
Z ∞
0 fE(n,n/t)(u)·FB(u)du, (145)
where fE(n,n/t)(u)is the density of an order-nErlang distribution with rate parameterν = n/tand we have thatFB(n)(t) → FB(t)asn → ∞. FB(n)(t)is basically the probability that the busy period is shorter than an Erlang(n,ν)variable.
Specifically for the busy period analysisFB(n)(t)can be obtained according to the next the-orem.
Theorem 20. ([72], Theorem 4) The order-napproximation of the busy period distribution is FB(n)(t) =κ
n−1 k
∑
=0Ψνk1, (146)
where matricesΨνkare defined recursively as
(Q+++Ψν0Q−+−νI)Ψνk+Ψνk(Q−−+Q−+Ψν0−νI)
=−2νΨνk−1−
k−1
∑
i=1ΨνiQ−+Ψνk−i, (147) fork>0, andΨν0is the solution to theNARE
ΨνQ Ψν+Ψν(Q −νI) + (Q −νI)Ψν+Q =0. (148)
4.2 markovian fluid models 69 For the detailed proof of the theorem, see [72]. The idea is to construct a special fluid model which counts the number of Exp(ν) events during the busy period. MatrixΨνk is the probability that k such events occur before the end of busy period (with the usual phase-transition probabilities being the entries of the matrix). If the number of Exp(ν) events is less thann, then the busy period is shorter than an Erlang(n,ν)variable, providing (145).
5
A N A LY S I S O F T H E M A P / M A P / 1 Q U E U E
The MAP/MAP/1 queue is aFCFSqueue where the arrivals of customers are given by aMAP characterized by matricesD0andD1, and the service process is described by aMAPas well, given by matricesS0andS1.
Thus, both the inter-arrival and the service times can be non-exponential and correlated.
The majority of queueing models consideriid.service times, which is a reasonable assumption in many practical systems. For this specific queue, however, modeling the service process by aMAPmakes the discussion simpler, and since thePHrenewal processes are the sub-classes ofMAPs, this choice makes the queueing model more general (it is described in Section3.1.1 how to representPHservice times with aMAP).
The performance measures in this system, including the queue length and the sojourn time distributions, can be derived by standard methods and are known for a couple of decades. Nev-ertheless, we are going to present them in Sections5.1and5.2as they provide an introduction to the apparatus applied for the analysis of more complex systems described in the subsequent chapters. In Section5.3.3several approximations for the departure process are discussed, in-cluding the joint moment based one, which plays an essential role in the queueing network analysis approach proposed in Chapter8.
5.1 analysis of the number of customers in the system
If we denote the mean arrival rate byλ = θAD11and the mean service rate byµ = θSS11 (withθAandθSbeing the stationary phase distributions of the arrival and service processes, respectively), the utilization of the queue is given byρ= λ/µ. In this chapter it is assumed that the system is stable, thusρ<1.
To analyze the number of customers in the system, aCTMCis created to model thequeue length process. While this seems to be a natural choice, it will be clear in Chapter6and7that in many systems this approach is either too complex or infeasible.
The CTMC characterizing the queue length process needs to keep track of 1)Y(t), the number of customers in the system, 2)JA(t), the phase of the arrival process, and 3)JS(t), the phase of the service process. By introducing the finite stateCTMCJ(t)as the direct product of JA(t)and JS(t), the queue length process leads to a homogeneous QBD (see Section 4.1.2), where the generator has a block tri-diagonal structure given by (117). The matrix blocks of the generator are defined by the following Kronecker operations:
F = D1⊗I, L= D0⊕S0, B= I⊗S1, L0= D0⊗I.
(149)
72 analysis of the map/map/1 qeue
The discussion of the basic properties of the Kronecker operations and how to use them to create the generator of independent, parallel Markov chains is provided in AppendixA.1.
The meaning of the Kronecker summation giving matrixLis that the arrival and the service MAPsare evolving in parallel moving along their internal transitions. The Kronecker product providingF (andB) are those transition rates of the arrival (and service)MAPsevolving in parallel that lead to arrivals (and services), respectively. Arrival events are accompanied by level forward and service events by level backward transitions in theQBD. At level0, where the system is empty, the serviceMAPgets frozen.
If the arrivalMAPhasNAphases and the serviceMAPhasNSphases, the cardinality of the superposed phase process (which is the size of the blocks of the generator) isN= NA·NS.
Let us denote the joint stationary distribution of the number of customers in the system and the phase process by vectoryi = {limt→∞P(Y(t) = i,J(t) = j),j = 1, . . . ,N}. According to (122)yihas a matrix-geometric distribution, thus
yi =y0Ri, i≥0. (150)
The details to obtain vectory0and matrixRare described in Section4.1.2.
The simplicity of the matrix-geometric distribution enables the efficient computation of many performance measures. E.g., thekth factorial moment of the number of customers in the systemE(Yk)can be computed as
E(Yk) =
∑
∞ i=0i(i−1)· · ·(i−k+1)y0Ri1= k!y0Rk(I−R)−(k+1)1, (151) and the generating function (GF)Y(z) =∑∞i=0ziyi1is
Y(z) =
∑
∞ i=0zi y0Ri1=y0(I−zR)−11. (152)
For the departure process analysis the distribution of the number of customers embedded just after the departures,xi, will be necessary. This distribution is computed by “weighting”
the elements of the stationary distribution with the transition rates leading to a departure, thus
xi = yi+1B
∑∞k=1ykB1 = 1
λyi+1B, i≥0. (153)
The normalization constant, the denominator of (153) is the mean departure intensity which equals the mean arrival intensityλwhen the queue is stable.