After having determined all the possible transition rates among system states, calculating the steady state distribution of the system is the straight way to achieve session blocking probabili-ties and channel utilization values. We assume Poissonian arrival of sessions and these type of arrivals have the well-known property of seeing the time average (PASTA property – Poisson Arrivals See Time Average), e.g. stationary distribution of the system ([98]). Thus determin-ing blockdetermin-ing probabilities means summdetermin-ing up the probabilities of those states in which session rejection may occur due to the lack of base station capacity.
To determine the steady state distribution of the system we must enumerate the states and compose the infinitesimal generator matrix of the system using the above determined possible transmissions and their rates. Given the infinitesimal generatorQ∗ (note that it is not the same as the infinitesimal generator of the Markov chain describing the variability of the transmission rate of a customer, formerly denoted byQ(k)), the steady state probability vectorp∗ is calculated by solving the well known set of global balance equations, i.e.
p∗Q∗ = 0 p∗h= 1.
Unfortunately in practical scenarios solving this system of equations is not possible because of the huge number of states. In very simple scenarios the state space may contain only a few hundred thousand states. In this case the system of equations may be solved by some numerical method (e.g. Gauss-Seidel). But in most cases the number of states easily exceeds tens of mil-lions, than solving the system of global balance equations is impossible. Moreover, if a session may arrive with zero transmission rate, the state-space becomes infinite, thus this approach is not applicable as well.
Fortunately, to determine blocking behavior and system utilization, we do not need the steady state distribution itself. Rather it is enough if the probability of havingmunits of system capac-ity occupied is known, m = 1, . . . , C0. These parameters are referred as channel occupancy probabilities in the remainder of this Chapter.
Markov chain. To do this, we consider the theoretical case when the base station has infinite capacity, i.e. C0 = ∞. Moreover, this case has not only theoretical, but practical significance in teletraffic modeling of systems, namely when system performance is determined in terms of overload probability, that is the probability of total occupied capacity exceeding a given value.
This approach is used in the literature when dealing with CDMA networks, where capacity does not have a hard bound, but overload can be well defined.
It is clear, that when the base station is supposed to have infinite capacity, the Markov chain has a product form solution. This is because the infinite capacity case is equivalent with a queue-ing network, if each phaseiof the service time distribution is modelled by a single infinite server exponential queue with rate−Siiand with routing probability −SSij
ii between queuesiandj. We know from the famous BCMP theorem [99], that this network has a product form equilibrium distribution, containing the product of the steady state distribution of individual queues. In the context of queueing networks, this is the usual use of the term product form solution. However, as the system under consideration in this thesis is a single queue we use the term in more general sense. That is: product form distribution is a multi-dimensional distribution that is the product of its marginal distributions (e.g. [100]). Naturally, this latter more general definition contains the stationary distributions of product form queueing networks, as the distribution of the queue length of individual queues are the marginal distributions of the joint queue lengths distribution.
The system’s analogy with the BCMP network containing IS queues also means that this system is quasi-reversible and local balance equations hold (e.g. [101]). We now that in BCMP networks the local balance equations equate the input and output rates of a network station.
Obviously, in our problem the non-blocking part of the state space (i.e. those states, where any state transition can take place, the idle capacity of the base station is enough to accommodate a new connection with any instantaneous transmission rate) is equivalent with the state space of the infinite capacity case, in terms of the possible state transitions and their rates. Thus, in the non-blocking part the local balance equations will also hold. We will use the local balance equations to determine the closed form equilibrium distribution (product form) and later use it for approximate analysis as well. The BCMP analogy means that the following transitions hold balance in our system:
• transitions that result in an increment of the number of customers receiving a particular phase of the service time, caused by an arrival of a new or handover session, or by the phase change of an active customer
• transitions that result in the diminution of the number of customers receiving the same phase of the service time, caused by one customer leaving the system (either by handover or by connection termination), or by phase change of a customer.
Since no transition is allowed among different customer types, the following equations are valid for any type k sessions initiated within the examined cell or arrived after handover. The following derivation considers typek new connections, but for the sake of better readability the corresponding part of the state vector, i.e. vector n(N,k) is simply denoted by n∗. Moreover, for better readability we omit the superscript (N, k) of the corresponding service time matrix and initial vector as well. Putting all these into the language of mathematical symbols, the local balance equations have the form:
λNαksip(n∗) +
P
X
j=1,j6=i
(n∗j + 1)Sji·p(n∗+ej) =
(n∗i + 1)·p(n∗+ei) Si0+
P
X
j=1,j6=i
Sij
! ,
(3.7)
whereei denotes a vector of lengthP filled with zeros and a1in itsith position and for the sake of simplicity the number of phases of the service time of typek new sessionsND,R(k) ·NL(k)·NQ(k) is denoted by P. Although these local balance equations are determined using the system’s identity with a BCMP network, an alternative proof of the validity of (3.7) equations can be found in Appendix C.
Rearranging (3.7) and writing all the local balance equations into a single vectorial equation, we get
−λNαks·p(n∗) = [(n∗1+ 1)p(n∗+e1), . . . ,(n∗i+ 1)p(n∗+ei), . . . ,(n∗P+ 1)p(n∗+eP)]S. (3.8) Introducing the vector
F =
(n∗1+ 1)p(n∗+e1)
p(n∗) , . . . ,(n∗P + 1)p(n∗+eP) p(n∗)
(3.9) from (3.8) we have
F =−λNαksS−1. (3.10)
We can see that the relation of the steady state probabilities of neighboring states is contained in this vector, namely
Fi·p(n∗) = (n∗i + 1)·p(n∗+ei). (3.11)
i,j i+1,j i-1,j
i,j+1
i,j-1 i+1,j-1
i-1,j+1
s1
λ⋅ s2
λ⋅
s2
λ⋅
s1
λ⋅
0
S1
i⋅ (i+1)⋅S10
0
S2
j⋅
0
) 2
1 (j+ ⋅S ) 21
1 (j+ ⋅S
S21
j⋅ (i+1)⋅S12 S12
i⋅
i,j i+1,j
i-1,j
i,j+1
i,j-1
F1
F2
F2
F1
i (i+1)
j ) 1 (j+
Figure 3.3: Two dimensional state space example
This indicates that the equilibrium probability of state n∗ in the infinite capacity case can be calculated recursively from state0, by means of iteratively substituting into (3.11). This results in the steady state probability ofn∗as
p(n∗) = p(0)·
P
Y
i=1
Fin∗i · 1
n∗i!. (3.12)
An alternative derivation of (3.12), based on the BCMP analogy can be found in Appendix C.
We observe that (3.12) is identical with the steady state distribution of a multiclassM/G/m/m or M/G/∞system (e.g. [102]), with P classes and with offer loads Fi for classi(no change of required capacity during a connection). Figure 3.3 shows an example of a two dimensional system, namely the state space around state(i, j). This state space would describe a system with single session type that has two possible transmission rates and exponential channel occupancy time. Only those transitions and states are plotted that affect state (i, j). The right hand side of the figure shows the equivalent state space: here the elements ofF replace the arrival rates, while the leaving rates are normalised to1. We see that with this description based on the local balance equations the ”crossing” transitions disappeared and their influence is included inF. As the elements ofF characterise the ratio of the stationary probabilities of two neighboring states (3.11), I use the notion of multiplier factor for the elements of F and multiplier vector forF. Now, returning to the original, complete notions of this thesis, the complete and correct form of the multiplier vector for a typeknew connection is:
F(N,k) =−λNαks(N,k)
S(N,k)−1
. (3.13)
The complete expression of the product form distribution defined in (3.12) is [99]
p(n) = 1 G
K
Y
k=1
P(N,k)
Y
i=1
Fi(N,k)n(N,k)i 1 n(N,k)i !
P(H,k)
Y
i=1
Fi(H,k)n(H,k)i 1 n(H,k)i !
(3.14)
whereP(N,k)denotes the number of phases of typek new customers’ service time andGis the normalization constant so that the sum of probabilities of all states equals to one. Namely
X
alln
p(n) = 1 ⇒ G=X
alln K
Y
k=1
P(N,k)
Y
i=1
Fi(N,k)n(N,k)i 1 n(N,k)i !
P(H,k)
Y
i=1
Fi(H,k)n(H,k)i 1 n(H,k)i !
.
(3.15) Recognizing that the infinite capacity assumption allows the sum to run from0to∞for allni-s, the sum is the Taylor series of the exponential function, namely
G= exp
K
X
k=1
(
P(N,k)
X
i=1
Fi(N,k)+
P(H,k)
X
i=1
Fi(H,k))
. (3.16)
In practical cases the number of phases of the service time distribution may be quite large, es-pecially if exact PH fitting with numerous phases (e.g. 10-12) is used for the session length and dwell time distributions and sophisticated user traffic model is used. In this case the matrix inversion of (3.13) is the most resource demanding task.