The above construction of the service process should be carried out for each customer type k, k∈ {1...K}. Moreover, handover and new connections of the same type are different in terms of their describing times and the initial probability vector of the traffic characterising Markov chain, thus each customer type requires the use of two service time distributions. This results in the composition of 2K service time distributions. In this proposed model we do not allow a customer to change its service type during a session, thus the service times could be handled independently.
steady state distribution exists. If zero transmission is allowed (e.g. the well known and widely used on-off sources) the state space becomes infinite. However, regardless a session’s transmis-sion state it stays in the system until the channel occupancy time and since no restrictions were given on the number of customers in the cell from this point of view the cell acts as an infinite server queue, thus the criteria for stability is to have incoming rates finite, that is completed in our case with the requisite of having finite rates of the traffic describing Markov chain.
To unambiguously describe the Markov chain representing the proposed cellular system, the possible state transitions and the rates of these transitions are needed. State transitions may occur because of the following events: a handover or new connection arrives to the system, a session leaves the system either by handover or by connection termination or a customer changes the phase of its service time, this latter obviously includes the change of transmission rate of a customer. To simplify notations in the following discussion we do not use the whole state vector n, rather its subvectorn(N,k)orn(H,k)that changes due to a state transition. The state transition rates are also dependent on the applied admission control policy of the base station described in Section 2.4.
Recalling Section 2.1 the incoming rate of handover and new connections is denoted byλH
andλN, the probability that a particular arriving customer is of typek is denoted byαk. If for typekadmission control policy1is applied the state transition rates are the following:
• state transition due to a new connection arrival: this event results in a state transition from staten(N,k)into staten(N,k)+eiat rateλN·αk·s(N,k)i , whereeiis a vector of the same size as the phase number of the service time of typek connections, filled with0’s and one1at itsith position;
• state transition due to handover arrival: this results in a state transition from staten(H,k) into staten(H,k)+ei at rateλH·αk·s(H,k)i ;
• state transition due to session termination: this event results in a transition from staten(N,k) into staten(N,k)−ei at raten(N,k)i ·Si(N,k,0), where the vectorS(N,k,0) contain the finishing rates from all the phases of the typeknew connection service time;
• state transition due to the phase change of the service time distribution: this results in a transition from staten(N,k)into staten(N,k)−ei+ej at raten(N,k)i ·Sij(N,k).
Obviously the above transitions may occur when the total occupied capacity after transition does not exceed the amount that is allowed for a particular connection type, namely in case of an
arrival with transmission ratec(k)i
Coc+c(k)i ≤Ck,N or Coc+c(k)i ≤Ck,H,
or in case of phase change that result in a change of transmission rate fromc(k)i toc(k)j Coc−c(k)i +c(k)j ≤Ck,N or Coc−c(k)i +c(k)j ≤Ck,H,
where againCoc denotes the amount of total occupied capacity prior to state transition.
To classify the transition rates when admission control policy2(burst CAC, rate reduction) is applied, we suppose without the loss of generality that the possible transmission rates (so the states of the traffic describing Markov chain as well) are enumerated in increasing order from the lowest rate to the highest. It is clear from the construction of the service time distribution that for those phasesiandj of the service time distribution that correspond to different states of the rate describing Markov chain, but to the same phase of the channel holding time distribution
|i−j|=l·ND(R,k)·NL(k) l ∈h
1, . . . , NQ(k)−1i
holds. In the case when policy2is applied and a connection is forced to reduce its transmission rate due to the lack of capacity, the initial phase of its service time is altered to a phase that corresponds to a lower transmission rate but to the same phase of the channel occupancy time.
Supposing that phase i of the service time distribution corresponds to the highest admissible transmission rate the state transition rates due to arrival has the form:
• a new or handover connection results in a state transition from staten(N,k) or n(H,k) into staten(N,k)+ei orn(H,k)+ei at rate
λN·αk·
NQ(k)−
$
j N(k)
Q
%
−1
X
l=0
s(N,k)
i+l·ND(R,k)·NL(k) and λH·αk·
NQ(k)−
$
j N(k)
Q
%
−1
X
l=0
s(H,k)
i+l·ND(k)·NL(R,k). By observing the difference between the incoming rates of the two investigated admission control policies it is clear that policy 2 allows more connections to be admitted in case of an overloaded base station. After all, although this policy generally reduces blocking probabilities, reducing the transmission rate of a connection may not be realized for some type of connections due to the resulting degradation of packet level QoS measures.
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.