In this section we give a brief description of the DTM algorithm as introduced by Brahimi [51].
Initially the general continuous service time distribution is discretised by rep-
resenting it by m equally spaced values: u,2u, . . . , mu, and by assigning a proper
probability to each of these values. In this way the service procedure is split into m stages. The time unit is set to be equal to the time interval of the basic service stage
service stages that each customer in service requires, we have a full description of the current system’s state and due to the markovian arrivals all possible system’s states which can be reached at the next epoch can be calculated.
This is done by first removing the service completions and updating the remaining
residual service times. The probabilities that we have 0,1,2, . . . , r arrivals in the unit
interval are then calculated (i.e. V0(t), V1(t), V2(t), . . .). The algorithm then expands
like a tree, having as a root the current state, and as branches the states that arise if we take into account the different possible arrivals. However, the new states are not specified completely. We also have to incorporate the new service times of customers who start service by taking into account all the possible combinations of service time demands that could occur. In this way we calculate iteratively the system’s state probability distribution at successive time points.
We will now demonstrate how the DTM algorithm applies for each time step. Let
us assume that we are at time t and we are trying to calculate the state probabilities
at time t+ 1. We consider one by one all the states at time t and we update them
as described above to calculate the probabilities of the resultant states that might occur. We will show how the algorithm works for one of them. Suppose we consider
the state [nt :x1, x2, . . . , xm].
• First we need to remove from the system customers that complete their service
before timet+1. Since there werex1customers at timetwith one remaining unit
of service, at time t+ 1 these customers will leave the system. Also customers
that were in service will now reduce their remaining service time by one unit.
For this reason the resultant state is [nt −x1 :x2, x3, . . . , xm,0].
• Then the number rof possible arrivals during (t, t+ 1) needs to be added in the
system so the resultant state is [nt+1 =nt −x1+r :x2, . . . , xm,0], wherer can
take any non-negative integer value.
• The number of free places in the service that can accept new customers is then
i∈ {1, . . . , m}. In this way we create vectors of new service times (z1, . . . , zm),
wherezi is the number of customers starting service who needistages of service.
For this reason newc = Pm
i=1zi. Each of these vectors z = (z1, . . . , zm) has
an associated probability, which can be calculated by taking into account all possible combinations of service requests. It is:
P(z) =prob(z1, . . . , zm) =
newc!
z1!. . . zm!
S(1)z1. . . S(m)zm
where S(i) is the probability that the service time will last i units of time.
• The final states are now of the form [nt+1 :x2+z1, . . . , xm+zm−1, zm] and the
probability of reaching each of these states is the probability of being initially
at state [nt :x1, . . . , xm], multiplied by the probability of r arrivals, multiplied
by the probability of having [z1, . . . , zm] new service demands. This is:
Pt+1[nt−x1+r:x2+z1, . . . , xm+zm−1, zm]=Pt[nt :x1, . . . , xm]×Vr(t)×P(z) (3.2)
According to this forward algorithm, starting from a specific state at time t,
Equation (3.2) gives its contribution to the probability of a resultant state at time
t+ 1. We should repeat this calculation for all possible resultant states. Then, by
sweeping all possible states at time t we can find all possible states at time t+ 1.
Each time a resulting state has an associated probability (i.e. was also resulting state from a previous initial state), the latest probability contribution is accumulated to the previous one. In this way at the end of this procedure we have the system’s state
probability distribution at timet+ 1. Having found the system’s state distribution at
timet+ 1 we use it as a starting point in order to find the system’s state distribution
3.10
Summary
In this chapter we have described the discrete-time modelling of queueing sys- tems. According to this method the description of the systems under consideration is expressed fully by a system of difference equations. We have reviewed the literature of the two major categories used to solve these equations, i.e. the numerical and the analytic methods. We have also described the DTM algorithm which we are going to use in this research. We are next going to investigate whether the DTM theory that
up to now has been used successfully to modelM(t)/G/s(t) systems, can be extended
Chapter 4
Extending the DTM theory to
include state-dependent balking
4.1
Introduction
The aim of this chapter is to develop the DTM theory described in chapter 3 in order to include balking. The rate at which balking occurs depends on the state of the system, i.e. on the number of customers in the system. In practical terms this corresponds to informing incoming calls about their expected waiting time, or their position in the queue, so depending on how long this is, they will either enter the system, or hang up immediately which is called balking. In this way, from a formulation point of view, balking is about introducing state-dependent entries in DTM.
The entry process is assumed to be a state-dependent Poisson process, in which the arrival rate changes when an arrival manages to join the system and when a departure occurs. This chapter considers the entry process in two stages. In Sections 4.2-4.5 theory is developed for Poisson processes where arrival rates only change due to arrivals, i.e. the effect of departures is ignored. Then in Section 4.6 two approximate methods for introducing departures are described.
When events occur as a Poisson process at constant rate, it is well known that
the arrival rate changes when arrivals occur, the number of arrivals during time T
will depend on the starting state x, and will not follow a Poisson distribution. In
section 4.2 these state-dependent entries are described while in section 4.3 a theorem is presented which allows their probabilities to be calculated. In section 4.4 another way of calculating these state-dependent entries is presented. This is motivated by the relevant literature and it leads to a different formula, however both formulae give the same results as expected. Neither of these formulae can deal with a recurrent arrival rate, thus section 4.5 extends the previous theorems to deal with this case.
Having calculated the state-dependent entry probabilities we want to incorpo- rate them in the DTM algorithm. This is done in section 4.6 by introducing two approximations.