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

_{opt.}

Open access peer-reviewed chapter

By Yury K. Belyaev and Asaf H. Hajiyev

Submitted: June 8th 2020Reviewed: September 16th 2020Published: October 29th 2020

DOI: 10.5772/intechopen.94066

Downloaded: 127

Various lifts’ systems with different control rules are considered. It is suggested to use the efficiency indexes: customer’s average waiting in lift cabin time and average total time, including the time of delivering the customer to the desired floor. Various control rules are introduced: Odd-Even, where one lift serves only customers in Odd floors and other lift only does that in Even floors Up-Down control rule where one lift serves only customers who are going from the first floor to the destination floor 2, 3,…, k; another lift serves customers from the first floor to the upper floor k + 1, k + 2, …, n. The results of simulation, allowing to compare various control rules relatively to the efficiency indexes, are given. It is introduced an optimal number of lifts, which minimizes number of lifts, minimizing a customer’s average waiting time. For some systems, the method of finding the optimal number of lifts, is suggested. Necessary figures demonstrating the operation of the lifts’ systems and the results of the simulation allow to estimate the efficiency indexes.

- simulation of various lifts’ systems
- odd-even
- up-down
- situation of control rules
- customer’s average waiting time and total service time

The world economy suffered a lot of losses after the coronavirus pandemic and it will take a long time for its rehabilitation. An important role in the development of the world economy will have the transportation and communication systems because it is necessary to renovate the economic communications among countries. For the investigation of the transportation and communication systems, mathematical models of queuing systems with moving servers are widely used. Typical examples of queues with moving servers are the lifts’ systems. Lifts and communication systems, the traffic, the airport and the shipping facilities have a lot of similarities. All of them are united by the same principle – servers are moving in these systems. Hence, mathematical models of lifts’ systems can be applied for the investigation of other systems with moving servers. As investigation of such systems by analytical approaches faces troubles, the use of the modern computers can allow to simulate their behavior. The simulation of such systems, various systems of programming (* Wolfram Mathematica*and others) allow to get close to reality, the numerical data of the desired parameters and give some advice for applications. The simulation

In this paper, the authors consider various lift systems with different parameters and different control rules. This paper can be regarded as a continuation of the authors’ investigations presented in [4, 5]. Hence, we follow the notations introduced in these papers.

Advertisement## 2. Control policies for the lifts’ systems

There are many various control rules for the lifts’ systems. We will consider only some of them, for instance, the * Odd-Even*system, where some lifts serve customers at the odd floors and other lifts, at the even floors. Another control rule, we call it following to [4], the

An interesting unofficial control policy was created in the seventy years of the XX^{th} century, by the students in the dormitory of the Moscow Lomonosov State University. There are 18 floors in the student dormitory and two lifts’ halls with four lifts in each. The first lift hall operates from the * 1st to the 12th, 14 *floors. In order that the lifts work more rapidly, it was skipped the odd numbered floors, after the 12th. There is also a second lift hall for serving on the 1st-10th floors. If in the first hall, a lift came to the first floor and the first student yelled the word

In [7], it was introduced the so called “situation control rule” for systems with two lifts. If both lifts are going from up to down, then all arrived customers (at the different floors) will be distributed between lifts. This control rule allows to exclude stopping both lifts almost at the same time, at the same floors. Such systems work effectively for high intensity of customers’ flows. For instance, if both lifts are going from up to down, then each lift system defines the floors where the lift must stop and serve the customers. In the case of a customer’s arrival at the new floor system, it must be recalculated the number of the floors where the lift must stop. Such a control rule allows using lifts capabilities in a uniform way. Although the “situation control rule” needs some additional software and technical equipment, nevertheless it improves the service (reducing customer’s waiting and service times), it saves energy expenses and increase the lifetime of the lifts.

Advertisement## 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;

* L*– is the systems with

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

* f*- is the floor of appearance of the

* f*- is the floor of destination of the

It is necessary to note that for some different * i*the

* t*- is the instant of appearance of the

* t*- is the instant of the beginning service of the

* t*- is the instant of end service of the

* t*- is the instant when lift on

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

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

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

* h*– time which is spent for opening and closing the floor’s door;

Usually, in practice, approximately * h*If we consider the stationary input flow, then, the following parameters are used:

* λ*–is the intensity of customers’ flow, which appears at the

* λ*=

_{2}= _{k1} – is the intensity of customers’ flow, which appears on the upper * {2, 3,…, n*floors, who want to go down to the first floor;

* CWT(S) – a customer’s average Waiting Time*in the system

* CST(S) – a customer’s average Service Time*in the system

* CTT(S) = CWT(S) + CST(S) – a customer’s average Total Time*in the system

For instance, * CTT(L*is a customer’s average total time, for a system in a building with

* 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

* T*) - cycle time of the

* T*(

* 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

We also introduce the new parameters for the lifts’ systems, which describe the * lift energy expenses*and the

* LEE*Average value of the

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

* SEE(S) –*average value of

_{1}(S) + LEE_{2}(S) + … + LEE_{n}(S);

* k*coefficient defining the lifts’ energy expenses, during a unit time, for opening and closing the doors;

* k*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 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

In the unloading regimes, when lifts are going from the upper * j*floor to the first floor, the lifts can take customers from

Remind that * L*is the system with

means that the lift is occupied;

means that the lift is empty (free);

means the instant of the customers’ arrival~~instant~~.

** Definition.**The flow of customers is called

Advertisement## 4. The systems L_{2}F_{n}C_{IL} and L_{2}F_{n}C_{DL} in the loading regimes

We will compare both systems L_{2}F_{n}C_{IL} and L_{2}F_{n}C_{DL} 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 * L*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

_{1} = t_{a}(1) = t_{a}(2)_{,} x_{2} = t_{b}(1) = t_{b}(2) = x_{1} + h_{d}, x_{3} = x_{2} + (f_{2}–1)h_{f}, x_{4} = t_{e}(1) = x_{3} + h_{d},

_{5} = x_{4} + (f_{3}-f_{2})h_{f},x_{6} = t_{e}(2) = x_{5} + h_{d}, x_{7} = t_{a}(3) = t_{a}(4) = t_{a}(5),

_{b}(3) = t_{b}(4) = t_{b}(5) = x7 + h_{d}, x_{9} = x_{8} + (f_{1}–1)h_{d}, x_{10} = x_{7} + (f_{3}–1)h_{f},

_{11} = t_{e}(3) = x_{9} + h_{d}, x_{12} = x_{11} + (f_{4}-f_{1})h_{f},x_{13} = t_{e}(4) = x_{12} + h_{d}, x_{14} = x_{13} + (f_{5}-f_{4})h_{f},

_{15} = t_{e}(5) = x_{14} + h_{d}, x_{16} = t_{a}(6), x_{17} = t_{b}(6) = x_{16} + h_{d}, x_{18} = x_{16} + (f_{5}–1)h_{f}.

Consider the system * L*with rare input flow in loading regime. Then, at the preceding instants of a customer’s arrival, both lifts occupy the floors

_{1} = t_{a}(1) = t_{a}(2)_{,} x_{2} = t_{b}(1) = t_{b}(2) = x_{1} + h_{d}, x_{3} = x_{2} + (f_{2}–1)h_{f}, x_{4} = t_{e}(1) = x_{3} + h_{d},

_{5} = x_{4} + (f_{3}-f_{2})h_{f}, x_{6} = t_{e}(2) = x_{5} + h_{d}, x_{7} = t_{a}(3) = t_{a}(4) = t_{a}(5),

_{8} = t_{b}(3) = t_{b}(4) = t_{b}(5) = x_{7} + h_{d},x_{9} = x_{8} + (f_{1}–1)h_{d},x_{10} = t_{e}(3) = x_{9} + h_{d}, x_{11} = x_{10} + (f_{4}-f_{1})h_{f},

_{12} = t_{e}(4) = x_{11} + h_{d}, x_{13} = x_{12} + (f_{5}-f_{4})h_{f}, x_{14} = t_{e}(5) = x_{13} + h_{d}, x_{15} = t_{a}(6), x_{16} = x_{15} + (f_{3}–1)h_{f},

_{17} = t_{b}(6) = x_{16} + h_{d}.

Thus, we have * CWT(L*and

* CWT(L*If an intensity of input flow is increasing, then the difference (

After some critical value of intensity * λ*this difference (

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 * L*and

** Remark.**For small values of intensity of the input flow, the system

* f*(

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

_{d} | _{f} | ||
---|---|---|---|

Simulation shows (see, Table 1) that typically.

2

Below, in the Table 2, the results of simulation for comparison of the systems * L*and

λ | _{2}F_{15}C_{IL}) | _{2}F_{15}C_{DL}) |
---|---|---|

0,009 | 29,34 | 44,24 |

0,012 | 29,63 | 44,33 |

0,015 | 44,32 | 44,63 |

0,018 | 47,17 | 46,37 |

0,021 | 59,29 | 51,09 |

0,024 | 72,27 | 63,57 |

0,027 | 86,85 | 76,65 |

0,031 | 94,02 | 79,12 |

0,034 | 103,44 | 85,24 |

0,037 | 118,21 | 99,31 |

0,040 | 132,65 | 120,55 |

0,043 | 152,04 | 145,04 |

_{2}* F*)

Consider the systems * L*and

The Table 2 and Figure 3 show that for a small intensity of the input flow of customers, we have * CTT(L* <

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

λ | _{2}F_{15}C_{DL}) | _{2}F_{15}C_{SC}) |
---|---|---|

0,009 | 44,24 | 44,24 |

0,012 | 44,33 | 44,21 |

0,015 | 44,63 | 44,65 |

0,018 | 46,37 | 45,15 |

0,021 | 51,09 | 47,61 |

0,024 | 63,57 | 55,41 |

0,027 | 76,65 | 66,06 |

0,031 | 79,12 | 64,22 |

0,034 | 85,24 | 69,94 |

0,037 | 99,31 | 79,47 |

0,040 | 120,55 | 98,34 |

0,043 | 145,04 | 120,63 |

Advertisement## 5. Situation control rule

Introducing the * SC*(situation control) allows to reduce the

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

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

λ | _{2}F_{15}C_{IL}) | _{2}F_{15}C_{DL}) | _{2}F_{15}C_{SC}) |
---|---|---|---|

0,009 | 29,34 | 44,24 | 44,24 |

0,012 | 29,63 | 44,33 | 44,21 |

0,015 | 44,32 | 44,63 | 44,65 |

0,018 | 47,17 | 46,37 | 45,15 |

0,021 | 59,29 | 51,09 | 47,61 |

0,024 | 72,27 | 63,57 | 55,41 |

0,027 | 86,85 | 76,65 | 66,06 |

0,031 | 94,02 | 79,12 | 64,22 |

0,034 | 103,44 | 85,24 | 69,94 |

0,037 | 118,21 | 99,31 | 79,47 |

0,040 | 132,65 | 120,55 | 98,34 |

0,043 | 152,04 | 145,04 | 120,63 |

Advertisement## 6. Energy expenses

Now we will show that introducing ~~of~~ the control rules, will be ~~to~~ reduced not only the * CWT*and the

* CTT(L* <

and moreover, from formula (1), it follows

* LEE(L*and

i.e. _{2}F_{n}C_{IL}) < LEE(L_{2}F_{n}C_{DL}).

Energy expenses linearly depend on the * CTT*and also on the

* CWT(L*and

_{2}F_{n}C_{IL}) = h_{f} (n-1) + 2h_{d}, LEE(L_{2}F_{n}C_{IL}) = k_{f}(n-1)h_{f} + 2k_{d} h_{d},

* SRT = k*Then, for Poisson flow of customers with intensity λ during the time interval

Advertisement## 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

The main efficient parameters of the lifts’ systems are * the customers’ average waiting times (CWT)*and

The estimates of the efficiency of the * CWT*and

Below, in Table 7, we illustrate the following control rule~~s~~: lift * L1*serves lower floors, from floor

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

In Table 8, two lifts can stop * L1,*on

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,

Advertisement## 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

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