**3. The mathematical models of the lift systems**

For constructing the mathematical models of the lifts'systems, we use conceptions and parameters introduced in [4, 5]. The followings notations are introduced:

*n* – is the number of the floors in the building;

*k* –is the number of the lifts in the building;

*LkFnCxx* – is the systems with *k* lifts, *n* floors and control policy *xx*;

*i* - is an ordered in time identifying number of a customer during simulation;

*fa (i)* - is the floor of appearance of the *i-th* customer;

*fd(i)*- is the floor of destination of the *i-th* customer.

It is necessary to note that for some different *i* the *fa (i)* and *fd(i)* can take the same value.

*ta (i)* - is the instant of appearance of the *i-th* customer;

*tb (i)* - is the instant of the beginning service of the *i-th* customer in lift cabin;

*te(i)* - is the instant of end service of the *i-th* customer;

*tc(j)*- is the instant when lift on *j-th* cycle is returning to the *1-st* floor;

*n* – number of the floors in the building;

*r* – roominess, restriction of maximum possible number of customers, who can be in the lift cabin;

*hf -* time necessary for the lift to move up or down, between two neighboring floors;

*hd*– time which is spent for opening and closing the floor's door;

Usually, in practice, approximately *hd = 2hf.* If we consider the stationary input flow, then, the following parameters are used:

*λf1f2* –is the intensity of customers' flow, which appears at the *f1-th* floor and want to go to *f2-th* floor;

*λ1*= P*<sup>n</sup> <sup>k</sup>*¼<sup>2</sup> *<sup>λ</sup>*1k - is the intensity of customers' flow, which appears at the first floor and are going to upper floors;

*λ*2= P*<sup>n</sup> <sup>k</sup>*¼<sup>2</sup> *<sup>λ</sup>*k1 – is the intensity of customers' flow, which appears on the upper *{2, 3, … , nf}* floors, who want to go down to the first floor;

*CWT(S) – a customer's average Waiting Time* in the system *S*, i.e. the mean time from the instant when a customer arrives at the system and waits until the instant when he gets the lift;

*CST(S) – a customer's average Service Time* in the system *S*, i.e. the mean time from the instant when the customer gets in the lift, until the instant when he gets off the lift;

*CTT(S) = CWT(S) + CST(S) – a customer's average Total Time* in the system *S,* which is measured as a mean time from the instant when the customer arrives into the system until he gets off the lift (arrival to ordered floor).

For instance, *CTT(LkFnCxx)* is a customer's average total time, for a system in a building with *k* lifts, *n* floors and control policy *xx.*

*IL*–independent lifts'system. It means that all the lifts are operating independently from each other, i.e. if at the preceding instant of a new customer's arrival, several lifts are free (empty), then, all of them will go to this customer's call. Such systems are often used in the buildings with two lifts.

*DL* – dependent lifts'system (for a customer' call, the nearest lifts going to him);

*UD(k)*- where one lift serves only customers who are going from the first floor to *2, 3, … , k;* and another one serves customers who are going from the first floor to upper *k + 1, k + 2, … , n;* In such systems, when an *Up* lift is going from *j1-th* floor (*j1 > k)* to down, it can take customers from *j2-th* floor *k < j2 < j1,* if there is an empty space in the cabin. Otherwise, the lift is directly going to the first floor. Similarly, when the *Do(wn)* lift is going from *j3-th* floor (*j3 < k)* to the first floor, it can collect customers from *j4 -th* floor *j4 < j3,* if there is empty space in the cabin.

*TU (L2FnCUD(k)*) - cycle time of the *Up* lift in the system *L2FnCUD(k); TD*(*L2FnCUD(k))*- cycle time of the *Do* lift in the system *L2FnCUD(k);*

*SC* – situation control - there is some (robot) software, which depends on new customers' arrivals, gives commands to the lifts where to stop and which floors to pass by. The appearance of a customer at the new floors can change the system of commands;

*LRC –*Average *Lift Return Cycle* time, i.e. the average time interval between two comings of the lift to at the first floor.

We also introduce the new parameters for the lifts'systems, which describe the *lift energy expenses* and the *single race time*:

*LEEj (S) –* Average value of the *j-th Lift Energy Expenses* in the system *S,* measured in *Kw* (kilowatt);

Note that Energy Expenses in *Kw* depend not only on the volume and weight of the cabin, but also on its speed, acceleration and deceleration. Empirically, electric Energy Expenses can be shown each day, on the electric counter of each lift.

*SRT(t)* – Average *Single Rate Time*, i.e. average time when the lift is moving without customers, during time *t*;

*SEE(S) –* average value of *System Energy Expenses*, i.e. average value of energy expenses of all the lifts in the system *(S).*

*SEE(S) = LEE1(S) + LEE2(S) + … + LEEn(S);*

*kd –* coefficient defining the lifts' energy expenses, during a unit time, for opening and closing the doors;

*kf –* coefficient defining the lifts' energy expenses, during a unit time, for covering the distance between two neighboring floors.

There are different regimes of operating the lifts'systems.

**Loading regimes**, where customers from the first floor are going to upper floors. Such regimes are observed in the office buildings, in the morning (08.00–09.30) when customers are going to their offices. Similar regimes are observed in the

*Estimation of the Efficiency Indices for Operating the Vertical Transportation Systems DOI: http://dx.doi.org/10.5772/intechopen.94066*

residence buildings, in the evening (17.30–19.00), when people come back home from their work.

**Unloading regimes,** in the office buildings, in the evening (17.00–18.00), customers stop working and go back by lifts, from their offices to the first floor.

There also exist **mixed regimes**, when customers from the first floor are going to the upper floors and vice versa. Moreover, there are customers who are going from *j1-th* floor to the *j2-th (j1, j2 = 2,3,..,n).* In this paper, only loading and unloading regimes will be considered. Some investigations of the mixed regimes can be found in [5].

In the unloading regimes, when lifts are going from the upper *j1-th* floor to the first floor, the lifts can take customers from *j2-th* floor (*j2* < *j1),* if there is a free space in the cabin. If, at some floor, the number of customers in the cabin became *r* (roominess), then the lift would go directly to the first floor, without stopping. This policy is observed in all the regimes.

Remind that *L2FnCn1,n2* is the system with *2* lifts, *n* floors and after completing the customer's service, one lift (with empty cabin) must go to *n1-th* floor, if there is no lift, otherwise, it should go to *n2-th* floor. Below, in the **Figures 1** and **2**, axes *x* means current time;

**Figure 1.** *Example of loading regime for* L2FnCIL*.*

**Figure 2.** *Example of loading regime for* L2FnCDL*.*

• means that the lift is empty (free);

• means the instant of the customers' arrival instant.

**Definition.** The flow of customers is called *rare* for the lift system *LkFnCxx*, if at the preceding instant of the customer's arrival, among the *k* lifts there is at least one (non-occupied) lift, which goes to the customer's call.
