# Applying of the queueing theory to model operating modes of a logging machine

## The results of the selected modes of operation of harvesting machines, by optimization of cut-to-length with application of methods of Queuing theory. Average cycle time for forest machines. The utilization ratio for different systems of forest machines.

Рубрика | Сельское, лесное хозяйство и землепользование |

Вид | статья |

Язык | английский |

Дата добавления | 29.04.2017 |

Размер файла | 557,4 K |

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Applying of the queueing theory to model operating modes of a logging machine

Shegelman Ilya Romanovich

Dr.Sci.Tech., professor

The results of operating conditions of harvesting forest machines by means of optimization harvesting forest process with application of the queueing theory are considered

Keywords: LOGGING, MODELING, WAITING THEORY, HARVESTER, FORWARDER

В статье рассмотрены результаты выбора режимов работы лесосечных машин путем оптимизации процесса заготовки сортиментов с применением методов теории очередей

Ключевые слова: ЗАГОТОВКА ЛЕСА, МОДЕЛИРОВАНИЕ, ТЕОРИЯ МАССОВОГО ОБСЛУЖИВАНИЯ, ХАРВЕСТЕР, ФОРВАРДЕР

The research of logging technology is conducted at Petrozavodsk State University, and it is reflected, for instance in works [3-5]. A goal of the research is to improve the technological processes of industrial and energy wood harvesting. This research has demonstrated a promising perspective of the queueing theory application for this purpose [1-2].

This approach has allowed studying the resulting effect caused by simultaneous utilization of a few harvesters and forwarders in the cutting area [6].

From the point of view of the queueing theory methodology [6], the technological timber harvesting process, using both harvesters and forwarders, may be considered as a queueing system with batch arrivals.

A harvester fells trees, cuts branches and bucks up on industrial and energy assortments. After that, a forwarder picks up assortments, loads them on its dray and then transports the assortments to a holding area.

A Job has assortments.

The harvester work can be fully described by the mean arrival rate which is the average number of the batches generating by the harvester *i* per unit of time. After the harvester produces assortments, then the job arrives to a factory. In general, the number is random and has an arbitrary distribution. If a workstation (forwarder) is busy then an arriving job goes to a waiting area. The waiting area is a collecting network. After the forwarder becomes free the job proceeds to process.

The forwarder work determines the mean processing rate (the average number of batches which any forwarder is processing per unit of time).

If there are m working harvesters then the mean arrival rate (the average number of the assortments which all harvesters are generating per unit of time) is the sum of the rates of m (independent) streams, that is

, (1)

where is the total number of the harvesters and is the mean arrival rate of harvester *i*. Values and are connected by the following expression , where is the mean number of assortments in a batch.

There are two the most important performance measures of the factory. It is the mean of cycle time and the mean of work-in-process . The mean of cycle time is the average time that the job spends within a system, and it includes the time which the job spends in а waiting area, and the time which the job is processed by the forwarder. For the queuing system the value is defined as follows:

, (2)

where is the mean time which the harvester processes an assortments batch; is the mean time which the job spends in a waiting area (in a queue); is the mean time which the job is processed by the forwarder.

The mean value can be obtained from the Wald's identity as follows

, (3)

where is the mean time between the moments of assortments' cut out (if all harvesters are taken into account).

The mean time which the job is staying in a waiting area of some forwarders is defined as

, (4)

where is the squared coefficient of variation of time which all harvesters process batch of assortments; is the squared coefficient of variation for the time which forwarder processes batch of assortments; is the utilization factor; is the total number of forwarders.

The utilization factor of forwarders is defined as . The value is defined by the formula

, (5)

where is the average number of batches generating by harvester *i* per unit of time, and it connects with the value by the formula ; is the squared coefficient of variation of the time which harvester *i* processes a batch of assortments. The value is defined as

, (6)

where is the squared coefficient of variation for the time which harvester *i* processes assortment; is the squared coefficient of variation for the total number assortments which are located on a dray of the forwarder.

The connection between the values and is expressed by the Little's law .

The reduction of can be achieved by the decreasing of time when the assortments are located on a cutting area. This may correspond to increased productivity and / or increase of the uniformity of the system if values and do not change.

We note that the utilization factor shows how much the forwarder is loaded. If we have one forwarder and , then the forwarder is overloaded. If then the (limiting) fraction of time the forwarder is free is . When the utilization factor approaches to 1 (from below) then the queue grows up nonlinearly. It leads to an overloading of the system and finally breaks up production.

At the time when the job is in a waiting area (in a queue) tends to 0 the system works more regularly.

We note that the mean work-in-process shows how many assortments are in the cutting area. Also we note that the considered forwarder is able to take 10 m^{3} of assortments on the dray. It means that one job is according to 10 m^{3} of logs.

To calculate the means and , a few experiments have been realized in Pryazha region, the republic of Karelia. The harvester John Deere 1270D Eco III and the forwarder John Deere 1110D Eco III worked in the cutting area. As a result, the following values , , , , , have been obtained.

, (7)

where is the *i-*th observation; is the absolute frequency; is the sample size; is the number of intervals. The empirical variance is defined as

. (8)

The squared coefficient of variation is defined by the formula

. (9)

Using formulas (1) - (6) and the model from the paper [1], we have obtained the following results:

Now we show how the system performance is changed depending on the changes of given parameters. We assume that the variances of the assortments' arriving time and the assortments' processing time decrease. It means that the machines are working more evenly. To realize it in the model, we must reduce values and . For example, if is reduced on 10 % then and are reduced on 0,31 %. In that case the utilization factor remains the same. The mean time the job is in a waiting area is reduced on 4,78 %.

If is reduced on 10 % then и are reduced on 0.03 %. (At that again the utilization factor is not changed.) The mean time the job is in a waiting area is reduced on 0.44 %.

If we increase on 10 % then the values and also increase on 0,31 %, while the value increases on 4,78 %.

If increases on 10 % then, as we can see, and increase as well. Moreover, the value increases on 0.44 %.

If increases on 10 % and is reduced at one time. Then the values and decrease on 0,28 %, and also decreases on 4,34 %. The utilization factor hasn't changed again.

If reduces on 10 % and increases at one time. Then the values and increase on 0,28 %, and the value also decreases on 4,34 %.

The small change of values and doesn't influence on the system.

If is increased on 10 % then increases on 3,7 % and reduces on 5,72 %. The value reduces on 26,73 %. The utilization factor is 0,66. It reduces on 9,09 %.

If is reduced on 10 % then is reduced on 1,73 % and is increased on 9,19 %, and the value is increased on 57,44 %. In that case the utilization factor increases on 11,11 % and becomes 0,81.

A change of may lead to a considerable change of and other parameters. In practice, the value depends on the construction of the harvester, skill of the driver, type of the cutting area, and other factors.

The value influences strongly on the mean cycle time and the mean of work-in-process . For example, if the value is reduced on 10 % then and are reduced on 6,25 %, is reduced on 35,96 % and is reduced on 10%.

If we increase on 10%, then and increase on 8,09 %, and , increases on 64,59 % and 10 %, respectively.

If is increased on 10 % and is reduced at one time then increases on 13,10 %, the value increases on 25,67 %, increases on 26,71 %, and utilization increases on 22,22 %. If is reduced on 10 % and is increased on 10 % at one time, then the reduction of the corresponding values are 1,75 %, 10,68 %, 50,29 %, 18,18 %, respectively.

We also can reduce and by changing the number of forwarders (and keeping rest of parameters, with exception of utilization factor, fixed). For instance, for , then the reduction of the values , , are respectively 5,99 %, 93,18 %, 50 % (with factor equals 0,36). For , values and reduce on 6,35 %, reduces on 98,76 %, and becomes 0,24 that is reduced on 66,67 %.

If we want to model the work of harvesters on the cutting area, we need to reduce the value twice.

Then, for instance, for , the value reduces on 30,75 % and reduces on 56,71 %, but the value increases on 38,49 %. At the same time, the utilization factor does not change. For , reduces on 33,12 % and reduces on 93,53 % but increases on 33,75 %. The utilization factor reduces on 33,33 % to be 0,48.

A complex of forest machines which has two harvesters and one forwarder is not used because the utilization factor is more than 1. It means that the mean time when the job is in a waiting area would increase with no limit.

Now we illustrate situation for harvesters on the cutting area. First, the value is reduced in 3 times. For this case and for the value reduces on 40,93 %, the value reduces on 74,44 %, while the value increases on 77,21 %. The utilization factor does not change.

If we continue to increase the number of machines, then we obtain the following. For 3 harvesters and 4 forwarders, the value decreases on 42,21 %, the value decreases on 94,44 % , but increases on 73,35 %. The utilization factor is reduced on 25 % and becomes equals 0,54. For 4 harvesters and 4 forwarders, we obtain reduction of and on 45,99 % and 82,77 %, respectively, while the quantity increases on 116,06 %. The utilization factor does not change.

The mean of cycle time, the mean time which the job is staying in a waiting area, the mean of work-in-process and the utilization factor for different complexes of forest machines are displayed in Fig. 1 - 4.

Fig 1. The mean of cycle time for different complexes of forest machines

Fig 2. The mean time which the job is staying in a waiting area for different complexes of forest machines

Fig 3. The mean of work-in-process for different complexes of forest machines

Fig 4. The utilization factor for different complexes of forest machines

Now we summarize our observations. If we use 1 harvester and 2 forwarders, then we obtain a considerable decrease of the mean time which the job is in the waiting area (93,18 %). However, the mean cycle time is reduced only on 5,99 % and the utilization factor is 0,36 so the forwarders will have too much free time. The increase of the number of forest machines increases the mean work-in-process. For example, assume that initial value = 1,84. It means that the cutting area has 18,4 m^{3} of the assortments. For 4 harvesters and 4 forwarders, =3,99. Assume, the cutting area has 39,9 m^{3} of assortments. The use of a large number of forest machines is difficult because in the case there appear problems with organization of working of several machines on the cutting area. In addition, the negative impact of machines on the forest increases.

On the basis of analysis of the model we can summarize that 2 harvesters and 2 forwarders is an optimal combination in conditions of the cutting area because in the situation we obtain considerable decrease of the mean cycle time and the mean job waiting time, while the utilization factor remains unchanged. timber harvesting optimization

*The present work is executed with financial support of the Ministry of Education of the Russian Federation within the limits of realization of the Program of strategic development of Petrozavodsk state university (PetrSU) for 2012-2016 «the University complex of PetrSU in scientifically-educational space of the European North: strategy of innovative development».*

Список литературы

1. Морозов Е. В. Вероятностно-статистический анализ процесса заготовки сортиментов / Е. В. Морозов, И. Р. Шегельман, П. В. Будник // Перспективы науки. 2011. №7(22). С. 183-186.

2. Морозов Е. В. О применении вероятностного моделирования для анализа некоторых технологических процессов лесозаготовок / Е. В. Морозов, И. Р. Шегельман // Глобальный научный потенциал. - 2011. - № 9. - С. 67-71.

3. Подготовка и переработка древесного сырья для получения щепы энергетического назначения (биотоплива) / И. Р. Шегельман, А. В. Кузнецов, П. В. Будник, В. Н. Баклагин, В. И. Скрыпник // Ученые записки ПетрГУ. Сер. «Естественные и технические науки». 2010. № 8(113). С. 79-82.

4. Шегельман И. Р. Классификация сквозных технологий заготовки биомассы дерева / И. Р. Шегельман, П. В. Будник // Перспективы науки - 2012. - № 4(31). - С. 90-92.

5. Шегельман И. Р. Способ выполнения лесосечных работ агрегатной машиной. Патент России № 2426303, МПК: A01G 23/00 / И. Р. Шегельман, В. И. Скрыпник, П. В. Будник, В. Н. Баклагин. Опубл.: 20.08.2011.

6. Curry G. L. Manufacturing systems modeling and Analysis / G. L. Curry, R. M. Feldman / Springer-Verlag Berlin Heidelberg, 2009. - 338 с.

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