**4. The systems** *L2FnCIL* **and** *L2FnCDL* **in the loading regimes**

We will compare both systems L2FnCIL and L2FnCDL with rare flow of customers, in the loading regime, relatively to a customer's waiting time (CWT). In the **Figures 1** and **2**, axes x means current time and axes y, an ordinal number of the floor, where the lift delivers the customers.

Below, in the **Figure 1**, the lifts' positions at the preceding instants of the customer's arrival are presented (rare flow) (see, **Figure 1**). If the input flow is rare, then, for the system *L2FnCIL* in loading regime, at the preceding instant of a customer's arrival one lift is located at the first floor and another is located at *j-th* floor, where *j = 2,3, … , n.* (see, **Figure 1**).

*x1 = ta(1) = ta(2), x2 = tb(1) = tb(2) = x1 + hd, x3 = x2 + (f2–1)hf, x4 = te(1) = x3 + hd, x5 = x4 + (f3-f2)hf,x6 = te(2) = x5 + hd, x7 = ta(3) = ta(4) = ta(5), x8 = tb(3) = tb(4) = tb(5) = x7 + hd, x9 = x8 + (f1–1)hd, x10 = x7 + (f3–1)hf, x11 = te(3) = x9 + hd, x12 = x11 + (f4-f1)hf,x13 = te(4) = x12 + hd, x14 = x13 + (f5-f4)hf, x15 = te(5) = x14 + hd, x16 = ta(6), x17 = tb(6) = x16 + hd, x18 = x16 + (f5–1)hf.*

Consider the system *L2FnCDL* with rare input flow in loading regime. Then, at the preceding instants of a customer's arrival, both lifts occupy the floors *2,3,..,n.* (see, **Figure 2**).

*x1 = ta(1) = ta(2), x2 = tb(1) = tb(2) = x1 + hd, x3 = x2 + (f2–1)hf, x4 = te(1) = x3 + hd, x5 = x4 + (f3-f2)hf, x6 = te(2) = x5 + hd, x7 = ta(3) = ta(4) = ta(5), x8 = tb(3) = tb(4) = tb(5) = x7 + hd,x9 = x8 + (f1–1)hd,x10 = te(3) = x9 + hd, x11 = x10 + (f4-f1)hf,*

*x12 = te(4) = x11 + hd, x13 = x12 + (f5-f4)hf, x14 = te(5) = x13 + hd, x15 = ta(6), x16 = x15 + (f3–1)hf,*

*x17 = tb(6) = x16 + hd.*

Thus, we have *CWT(L2FnCIL)=hd* and *CWT(L2FnCDL) = nhf/6+ hd*(1),

*CWT(L2FnCIL) < CWT(L2FnCDL).* If an intensity of input flow is increasing, then the difference (*CWT(L2FnCIL)*- *CWT(L2FnCDL))* is decreasing and goes to zero.

After some critical value of intensity *λ \** this difference (*CWT(L2FnCIL)* - *CWT (L2FnCDL))* is increasing until to some other value of intensity *λ\*\*.*

Afterward, it is again decreasing and goes to zero, for a high value of intensity. It is clear, that for a high intensity of the input flow, an operating of the systems *L2FnCIL* and *L2FnCDL* is becoming close to each other (see, **Figure 3**). In the **Figures 3***–***5**, axes *x* means intensity of the input flow and axes *y* means the value of the *CTT.* If roominess of the lift is bounded, then for a high intensity of the input flow it is not necessary to introduce any control, because both lifts stop at each floor and the system is operating like deterministic (at each floor there is always at least one customer).

**Remark.** For small values of intensity of the input flow, the system *L2FnCIL* is preferable than the system *L2FnCDL,* i.e. *CTT(L2FnCIL)* < *CTT(L2FnCDL).* There exists some interval (λ\* ,λ\*\*) of intensity which can be calculated by simulation),

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

**Figure 3.** *Graphs of the* CTT(L2F15CIL) *and* CTT(L2F15CDL).

**Figure 4.** *The graphs of the* CTT(L2F15CDL) *and* CTT(L2F15CSC).

**Figure 5**

*.*

 *The graphs of the* CTT(L2F15CIL), CTT(L2F15CDL) *and* CTT(L2F15CSC).

where the system *L2FnCDL* is preferable than the system *L2FnCIL*, because *CWT (L2FnCDL)* < *CWT(L2FnCIL).* Consider the system *L2FnCUD(k)*, where one of the two lifts, let us call it *Do-lift,* serves customers who are going from the first floor to *2,3,..,k* floors. Another lift, let us call it *Up-lift,* serves customers from the first floor to the upper floors *k + 1, k + 2,..,n.* Remind that *TU(L2FnCUD(k))* is the cycle time for the *Up*-lift, and *TD(L2FnCUD(k))* is the cycle time for the *Do*-*lift.* The cycle time of a lift is defined as the time interval between two sequential comings of the lift to the first floor. For this system we also introduce the floor number *fopt.* (optimal border cut), which can be found from the equation, when the cycle time of the *Up*-lift

closes to the cycle time of the *Do*-lift. In other words, *fopt.* is found from the following ratio.

*fopt. = {k: min/T<sup>U</sup>*(*L*2*FnCUD(k)*) -*TD*(*L*2*FnCUD(k)*)/}*k*

where **/./**means the absolute value of (.)*.* Below, as the result of the simulation, various systems are given. In the **Table 1**, for different number of the floors *(n),* the value of *fopt.* is given*.* For simulation, there were used the following lifts' parameters *hf = 4, hd = 7(Sec.).*

Simulation shows (see, **Table 1**) that typically.

2*n/3*≤ *f opt:* ≤ *3n/4*

Below, in the **Table 2**, the results of simulation for comparison of the systems *L2FnCIL* and *L2FnCOE* relatively to the *CTT,* are given. In the **Figure 3**, the graphical behavior of the *CTT* for both systems *L2FnCIL* and *L2FnCOE* is given. The results of simulation show that relatively to the *CTT,* the system *L2FnCOE* is preferable, than the system *L*2*FnCIL.*

*CTT(L*2*FnCOE*) ≤ *CTT*(*L*2*FnCIL*)) (see, **Table 2** and **Figure 3**).

Consider the systems *L2F15CIL*and *L2F15CDL.* It is necessary to note denote that by introducing the control rules, can be reduced not only the *CWT* (customer's waiting time) but also the *CST* (customer's service time) and hence, the *CTT* (customer's total time). Below, we will consider the *CTT* for all the systems.


#### **Table 1.**

*The values of* fopt. *For buildings with various floors*.


#### **Table 2.**

*The values of the* CTT (L2F15CIL) *and CTT (L2F15CDL).*


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

#### **Table 3.**

*The values of the* CTT(L2F15CIL) *and* CTT(L2F15CDL).


#### **Table 4.**

*The values of the* CTT(L2F15CDL*) and* CTT(L2F15CSC).

The results of simulation (see, **Table 2**) and the graphical behavior (see, **Figure 3**) of the *CTT* (customer's total time) are presented.

The **Table 2** and **Figure 3** show that for a small intensity of the input flow of customers, we have *CTT(L2F15CIL)* < *CTT(L2F15CDL)).* It follows from the fact that for a small intensity of the input flow at the preceding instant of a customer's arrival in the system *L2F15CIL,*one lift occupies the first flow and another one, *j-th* floor (j = 2,3,..,n). In the system *L2F15CDL,* for a small intensity at the preceding instant of a customer's arrival, both lifts occupy the first floor and hence, an average distance from lifts' position to customer call, far than in the system *L2F15CIL* (**Table 3**).

In the **Table 3** and **Figure 3**, the values of the *CTT*, depending on the intensity of the input flow for various systems, are shown. For a high intensity of the input flow, a difference between *CTT*(*L*2*FnCIL*) and *CTT*(*L*2*FnCOE*) is increasing, when the intensity of the input flow goes up (see, **Figure 3**), because in this case all the lifts stop at each floor and the system *L2FnCIL* operates like a system with one lift


#### **Table 5.**

*The values of the* CTT(L2F15CIL), CTT(L2F15CDL) *and CTT(L2F15CSC).*

(*L*1*FnCIL*) but with double roominess (see. **Figure 3**). Consider the systems with two lifts, with a situation control rule and denote it *L2FnCSC*, where the *SC* means situation control. At each given time-unit, the software is checking a new customer's arrival into the system and depending on this, each lift gets command, at which floor to stop for the customer's service. Below, in the **Table 1**, the result of simulation of such a system, for a building with 15 floors in an unloading regime, is presented. It is assumed that the roominess of the lifts is quite large and the lifts can take all the customers waiting on the floor. In simulation, we take *hf = 4, hd = 7.* Below, the results of the simulation (see, **Table 4**) and the graphical behavior (see, **Figure 3**) of the *CTT* (customer's total time), are presented. It is necessary to note that introducing of the control rules, can be reduced not only the *CWT* (customer's waiting time), but also the *CST* (customer's service time) and hence, the *CTT* (customer's total time).

### **5. Situation control rule**

Introducing the *SC* (situation control) allows to reduce the *CTT* for a high intensity of the input flow. Below, the results of the simulation (see, **Table 4**) and the graphical behavior of the *CTT(L2F15CDL)* and *CTT(L2F15CSC)* (see, **Figure 4**), are presented:

Data of the **Table 4** show that by increasing of the intensity of the input flow, the gain in the *CTT* is going up. In **Figure 4**, there are given the results of the simulation for the systems *L2F15CDL* and *L2F15CSC.* It is clear that for small and high values of intensity, it is not necessary to introduce the situation control, because both systems are almost the same and moreover, for high values of intensity, they coincide and the efficiency indexes can be calculated. There exists some interval where a difference between efficiency indexes takes maximal value and afterward it goes to zero, because for high values of customers' intensity flows, the lifts must stop at each floor, hence both systems have the same behavior (see **Figure 4**).

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

Below, the results of simulation (see, **Table 5**) and graphical behavior (see, **Figure 5**) of the *CTT(customer's total time)* for all the three systems, are presented. It is necessary to note that by introducing the control rules, can be reduced not only the *CWT (customer's waiting time*) but also the *CST(customer's service time*) and hence, the *CTT(customer's total time).*

## **6. Energy expenses**

Now we will show that introducing of the control rules, will be to reduced not only the *CWT* and the *CTT,* but also the *LEE* (*lift energy expenses*). Note, as it was mentioned above, for rare input flows it is not necessary to introduce the control rule *DL*, because.

*CTT(L2FnCIL)* < *CTT(L2FnCDL)* and moreover, from formula (1), it follows *LEE(L2FnCIL)=kdhd* and *LEE(L2FnCDL)=kf nhf/6+ kdhd* i.e. *LEE(L2FnCIL) < LEE(L2FnCDL).*

Energy expenses linearly depend on the *CTT* and also on the *SRT* (single rate time). As it follows from **Table 2**, the introduction of the SC (situation control) reduces the value of the *CTT,* by up to 25%. In [4] it was shown that for the *CTT (L2FnCIL)* in an unloading regime and rare flow of customers, the following ratio is true:

*CWT(L2FnCIL)=hf (n-1)/2 + hd, CST(L2FnCIL)=hf(n-1)/2 + hd* and *CTT(L2FnCIL)=hf (n-1) + 2hd, LEE(L2FnCIL)=kf(n-1)hf + 2kd hd,*

*SRT = kf hf [3(n-1)/4] + m kdhd.* Then, for Poisson flow of customers with intensity λ during the time interval *[0,t),* we have *SRT(t) =* λ*(kf hf 3(n-1)/4 + mkd hd)t.* As *λT* is an average number of arrivals during the time *T*, then *λTkf hf (n-1)* is an average energy which lift spends for serving the customers (motion of lift), during time *T.* As at each arrival instant, there is an average number of customers equal to *m,* then *λTm* is the average number of customers who arrived during the time *T,* into the system. For each customer's arrival, the lift spends the time *hd* for opening and closing the door. If we assume that each customer spends the time *hc* coming in and getting off a lift, then, the *mhc* is the time, which was spent for the *m* customers (coming in and getting off). Hence, a customer average energy spent for opening and closing the door, for customers coming in the lift and getting off equals

*2λ1Tkdhd + 2λ1Tmhc = 2λ1T(kdhd + mhc).* Thus, we have *LEE(L1FnCIL) = λTkf hf (n-1) + 2λT(kdhd + mhc).* Below, for simplicity, we assume *hc = 0,* which means that during the time *hd*, all the customers, who want to come in and get off a lift, can do it.

#### **7. Analysis of the two lifts system in planning office buildings**

Suppose it is a plan to construct a 15 floors office building with two similar lifts, which will carry in the morning, the customers to their offices and back, to the 1st floor, at the end of their work. It is necessary to introduce parameters of the lifts, e.g. roominess, velocity of lifts going up and down between floors and times for opening the doors on floors. Here, we consider unloading regime, where all the customers leave offices at the end of work hours. The offices will be placed on the floors *{2, 3, … ,15}.* The number of customers working on these floors will be *{12, 12, 15, 16, 12, 10, 17, 12, 14, 14, 16, 11, 18, 14}* i.e. all together in the building will be *nc = 193* customers. They should leave their offices during an interval of *1800 sec* (half an hour) in the evenings. The probability density *p[s] = 2(1800-s) /1800<sup>2</sup> , 0 < s < 1800* of customers to leave their offices is given in **Figure 6**:

**Figure 6.**

*The decreasing probability density* p[s] *during an interval of* 1800 Sec*.*

The main efficient parameters of the lifts'systems are *the customers' average waiting times (CWT)* and *customers' average total times (CTT).* Remind that the *CWT* is defined as an average time from the instant when the customer presses the button at the unloading, to get the lift cabin. *The CTT* is defined as the sum of the *CWT* and *CST,* i.e*.* the average time from the instant when a customer arrives into the system until the instant when it he has left the cabin, at the desired floor. Simulated data can be used to obtain estimates of the *CWT, CST* and *CTT.* The results of the lifts which are operating, can be described by initial histories of customers *{i, fa, ta, fd}, i = 1, 2, … ,nc*, where *nc* is the total number of customers in the building, *i* is the ordinal number of customer, *fd -* destination floor. For simulating an unloading regime, it is assumed that *fd = 1* and the lift spends *hf = 2.5* sec. to cross the distance between two neighboring floors. If the lift stops at some floor, then the time for opening and closing the door is *hd = 5 sec.* Let's assume that only one lift with roominess *r = 10*or*r = 20,* is operating. If the lift is located at the first floor (*fd = 1*) and its cabin is empty, then it immediately goes up to the highest *15th* floor. It means that we consider an unloading regime, where the lift is going from up to down and collects customers at the lower floors, if roominess allows it. Using our simulated program for unloading regimes, we have obtained six sets with initial customer histories, *{i, ta, fa}*: three the sets with roominess *r = 10* and three the sets with roominess *r = 20.*Using programming *Wolfram Mathematica*, we created the program, which transforms initial customer histories *{i, ta, fd}* of *nc* customers, into full histories *{i, fa, ta, tb, te}.*Here *tb* is the instant when the *i-th* customer goes in the lift cabin and *te* is the instant when the customer leaves the cabin at the *1st* floor. For simulating the 3-dimensional vector *{i,ta,fa},* the similar program has been created for loading regime. This program was used by comparing the full histories for a lift with *r = 10* and *r = 20*.

The estimates of the efficiency of the *CWT* and *CTT* are given in **Table 6**. It follows from **Table 6**, that roominess is the very essential parameter and the *CWT*


**Table 6.**

*The values estimation of the* CWT *and* CTT *obtained during the simulation unloading, three times (days), different lifts, with roominess* r = 10, 20*.*

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


**Table 7.**

*The values estimation of the* CWT *and* CTT *obtained for lifts L1 and L2, for data three "days".*


#### **Table 8.**

*The estimated values of the* CWT *and* CTT *for lifts* L1 *and* L2, *for data three "days".*

and *CTT* are better if *r = 20.* From the engineering point of view, it is more practical to use a lift system with two different lifts *L1* and *L2,* with roominess *r = 10* each.

Below, in **Table 7**, we illustrate the following control rules: lift *L1* serves lower floors, from floor *2* up to floor *10* and lift *L2* serves all the floors, from floor *11* up to floor *15.*

Note that we obtained in **Table 7**, better parameters, *for three days and two lifts,* than in **Table 6**. The above-considered data, for the *CWT* and *CTT*, correspond to three days. Note that our programs can simulate lifts for many days' operating data.

In **Table 8**, two lifts can stop *L1,* on *{1, 3, 5, 7, 9, 11, 13, 15}-* odd floors, *L2,* on *{1,2, 4, 6, 8, 10, 12, 14}-* even floors, and both lifts have *r = 10.*

We introduce the lifts'systems dispatcher (computer with special control lifts programs), as controller of the traffic of the moving lifts. Then, we can consider essentially many types of control rules for the lifts. For example, we can consider a system with two dependent similar lifts. They can stop, if their cabin contains less than r customers, follow specific rules at the floors with *waiting* customers, and if the system's dispatcher allows it.

## **8. Conclusion**

Several mathematical models of lifts'systems, which have different control rules, are introduced and investigated. By simulation, the data customer's waiting time *CWT* and total time *CTT* were estimated under different control rules and they have been compared relatively to the efficiency indices for the introduced control rules. The result of the calculation shows that relatively to the *CTT* usage of the situation control *SC,* in comparison with *DL* control rule, a gain of around 25% is achieved. If the roominess of the lift cabin is unbounded, then, for a high intensity of the input flow, it is not necessary to introduce any control. Then, it follows that if at each floor there is at least one waiting customer and both lifts stop at each floor, the system is operating like deterministic. The simulation also shows that, in the case of two lifts and a rare customers' flow, it is not necessary to introduce *DL* or *SC* control rules, because the system *IL* (with independent lifts) is preferable than the *DL* and *SC* control rules. It was shown that for a high value of an input flow of customers, the introduced control rule also reduces energy expenses, even by 25%, in some cases*,* which confirms the advisability of the introduced control rules. These results allow to make practical recommendations for reducing the various characteristics of the lifts'systems, such as the *CWT, CTT, SRT* (single rate time) and *LEE* (lift energy expenses). We completed the paper by examples with the calculation of a customer's waiting time *CWT* and a customer's average total time *CTT* for customers after work, for non-stationary cases, when there is an intensity of the customers' flow. It can be used for planning of the construction of the new office buildings with two similar lifts. The program can be extended for the case of several (more than two) lifts. We would like to underline that for the simulation of non-stationary cases, it is necessary to prepare a special program, which has a more complicated structure. In **Tables 6***–***8**, the results of the simulation for nonstationary cases, are given. We used the programming system *Wolfram Mathematica,* to create the programs for the simulation data and for a possible operation of two lifts. The results show that using simulation can help to estimate the appropriate values of roominess and find the optimal control rules, which can optimize the choice of the lifts' parameters (customer's waiting time, energy expenses and others). It can help for planning high floors buildings and future lifts' systems.
