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Calculation and analysis of network diagrams. Early date of the event

Basic network diagram parameters

The main parameters of the network diagram include:

Critical path

Time reserves for events

Time reserves for completing work

Path – a sequence of jobs in which the final event of one job coincides with the initial event of another.

Full path – a path, the beginning of which is the initial event, and the end of which is the final event.

The duration, the length of the path, is equal to the sum of the durations of the work. Its components.

Critical path – full path. the longest in duration of all paths in the network diagram from the initial event (I) to the final one (C).

The length of the critical path determines the total duration of the entire work package. The critical path allows you to find the timing of the final event.

Complete paths can pass outside the critical path or partially coincide with it. These shorter journeys are called relaxed. Their features are: That they have time reserves. But the critical path is not. For each i-th event the following is determined:

tpi– early onset– the minimum possible time for the occurrence of this event for a given duration of work.

t p i– late onset– the maximum time period for the occurrence of a given event, at which it is still possible to perform all the following work, in compliance with the established time period for the occurrence of the event.

R i – reserve time for event– the period of time by which the onset of this event can be delayed without disrupting the development period of the planned complex as a whole. Defined as the difference between the late ( t p i) and early ( t r i) the timing of the event.

Reserves for a critical path event are equal to zero, since on it t p i =t p i

For each job ( t ij) is determined:

early start date (t р.н. ij)– the minimum possible start date for this work.

early end date (t p.o. ij)– the minimum possible completion date for this work, for a given duration of work

late start date (t bp ij)– the maximum allowable start date for this work

late end date (t p.o. ij)– the maximum permissible deadline for completing this work, at which it is still possible to perform the following works in compliance with the established deadline for the completion event.

Obviously, the early start date of a job coincides with the early start date of its initial event, and the early finish date exceeds it by the duration of the job:

t р.н. ij = t r i

t p.o. ij = t r i + t ij

The late finish date of a job coincides with the late date of its end event, and the late start date of a job is less than the duration of the job:

t p.o. ij = t p j

t p.n. ij = t p j – t ij

Full reserve of time to complete the work R nij– the maximum period of time by which the start can be delayed or the duration of work can be increased without changing the established deadline for the completion event.

Free time reserve for completing work, which is part of the full reserve - the maximum period of time by which the start of work can be delayed or the duration of work can be increased without changing the early start dates for subsequent work.

Activities lying on the critical path have no reserves, since all reserves are created due to the differences in the durations of the critical and considered paths.

A relative indicator characterizing the time reserve for performing work is their tension coefficient, which is equal to the ratio of the duration of path segments between the same events, moreover, one segment is part of the path of maximum duration of all paths passing through a given work, and the other segment is part of the critical path.

3.Calculation of network models

Network parameters for network diagrams are calculated by graphical and tabular methods, and for complex ones by mathematical methods.

Graphically, the calculation method is carried out directly on the graph and is used in cases where the number of events is small. To do this, each circle is divided into 4 sectors.

Upper sector – reserve time for the event to occur R i

left sector – early date of event occurrence tpi

right sector – late date of occurrence of event t p i

below – event number

Parameter calculation method

1) Early timing of events . The early date of completion of the initial (first or zero) event is assumed to be zero. The early dates for the completion of all other events are determined in strict sequence according to increasing event numbers. To determine the early completion date of any event j, all work included in this event is considered; for each task, the early completion date of the final event is determined as the sum of the early completion date of the initial work event and the duration of this work t ij , From the obtained values, the maximum early time of the j-th event is selected

t pj = (t pi +t ij) max and is recorded on the graph (left sector of the event)

2) Late timing of events . The late date of completion of the final event is assumed to be equal to its early date. Calculation of the latest dates for the completion of all other events is carried out in reverse order, according to descending event numbers. To determine the late date for the completion of the previous event i, all work resulting from the i-th event is considered. For each job, the late date of completion of the initial event is calculated t p i, as the difference between the late date of completion of the final event of this work t p j and duration of this work t ij.From the obtained value, select the minimum time of the late completion date of the i-th event: t p i = (t p j - t ij)min and is recorded in the right sector.

3) Critical path duration equal to the early date of the completion event.

4) Event time reserves . When determining time reserves for events, you should subtract the number written in the left sector from the number written in the right sector of the given event and put it in the upper sector.

5) When determining the total reserve time for work, you should subtract from the number written in the right sector of the final event, the number written in the left sector of the initial event, and the duration of the work itself.

6) When determining the free reserve for work, you should subtract from the number written in the left sector of the final event, the number written in the left sector of the initial event, and the duration of the work itself.

Initial data:

Table method

Job codes in the table are written in ascending index order i.

Columns 2 and 3 are filled with auxiliary data: codes of previous and subsequent work. This data will be needed for calculations. If the work is initial, that is, there are no previous works, or final, that is, there are no subsequent works, then dashes are placed in the corresponding columns. There can be several preceding and subsequent works in accordance with the number of vectors ending or starting in a given event./

Column 4 contains the work duration values.

The calculated data begins in column 5. The calculation is performed in two passes through the rows of the table. The first pass along the rows from top to bottom, in which the early deadlines of the work are calculated, and the second pass along the rows from the bottom up, in which the late deadlines of the work are calculated.

The early start of work that has no previous ones (in column 2 - a dash) can be taken as 0, unless any other value is specified. The early completion of work is determined according to the formula t p.o. ij = t pH ij + t ij and is recorded in column 6.

The early start of the rest can be defined as, if, for example, work 2.5 is considered, which has an initial event of 2, then the time of its early start is equal to the time of the early end of work 12, since it has an end event of 2. The value from column 6 is rewritten to column 5 Codes of previous work are indicated in column 2. Early completion is also determined by the formula t p.o. ij = t pH ij + t ij

If, in column 2, it is indicated that a certain job is preceded by more than one job (jobs 5,6 are preceded by jobs 2,5 and 3,5), then you must select the early start value from several value options (9 - according to the end time of job 2 .5 or 13 – according to the time of completion of work 3.5). The selection rule corresponds to the formula t p .n. ij = (t pi +t ij) max , that is, the maximum value is selected (in the example - 16). Early endings are defined as above.

The maximum value of early termination in column 6 corresponds to the value of the duration of the critical path (16).

A second pass along the rows of the table from the work recorded in the last row to the work recorded in the first row allows you to determine the values of the later indicators of the activities. For jobs that do not have subsequent jobs (in column 3 there is a dash, in the example of jobs 46, 5,6), the value of the critical path is written in the late completion column (8). For these jobs, the late start value is calculated using the formula t p.n. ij t by ij - t ij

The late finishing of the rest can be determined as, if, for example, work 3.5 is considered, which has an end event of 5, then the time of its late finish is equal to the time of the late start of work 5,6, since it has an end event of 5. The value from column 7 is rewritten into column 8. Codes for subsequent work are indicated in column 3. Late start is also determined by the formula t p.n. ij t by ij - t ij .

If, in column 3, it is indicated that a certain job is followed by more than one job (work 0,1 is followed by jobs 1,2 and 1,3), then you must select the late finishing value from several value options (3 - according to the start time of work 1 ,3 or 7 – according to the start time of work 1,2), the minimum value is selected (in the example – 3). Late onset is determined as indicated above by the formula t p.n. ij t by ij - t ij .

The value of the total slack time (column 9) is calculated using the formula

R nij = t by ij - t pH ij - t ij.

The value of free time reserve (column 10) is calculated using the formula

R с ij = t ро ij - t рр ij - t ij

Any sequence of activities on a network schedule in which the ending event of each activity coincides with the starting event of the activity following it is called by.

A network path in which the starting point coincides with the initial event, and the ending point coincides with the ending event, is called full.

Path from the original event to any taken preceded to this event. The longest path preceding an event is called maximum previous. It is denoted L 1 (i), and its duration is t.

The path connecting any taken event to the final one is called subsequent way. The path with the longest length is called maximum subsequent and is denoted by L 2 (i), and its duration is t.

The complete path having the greatest length is called critical. Paths other than the critical path are called relaxed. They have time reserves.

Critical path activities are highlighted with bold or double lines. The duration of the critical path is considered the main parameter of the schedule.

Let's consider an algorithm for determining the critical path on a network diagram using the algorithm of the dynamic programming method.

Let's arrange the vertices of the graph by rank and number them from end to beginning. This will make it possible to combine the rank numbers with the stages of backward movement when finding conditionally optimal controls on the last one, the last two, etc. stages. Let's look at finding the critical path using the example of the network diagram shown in Fig. 10.7.

According to Bellman's principle of optimality, optimal control at each stage is determined by the control goal and the state at the beginning of the stage. The state of the system is the events lying on the ranks. To accomplish the final event X 16, it is necessary to complete the preceding events. Possible states of the system at the beginning of the last stage of work are the completion of events X 14 and X 15. In the circles at points X 14 and X 15 we put the maximum duration of work at the last stage: X 14 5, X 15 7. Let's find the maximum duration of work in the last two stages. The state of the system at the beginning of the penultimate stage is determined by the event X 13. The maximum duration of the path leading from X 13 to X 16 is equal to .

Therefore, in the circle next to the event X 13 you need to put the number 14, etc. Carrying out the stages from end to beginning, we find out the length of the critical path tcr =96. To find the critical path itself, we will go through the calculation process from the initial event X 1 to the final event X 16. We got the number 96 at the first stage (from the beginning) by adding 16 to the number 80. Therefore, the critical path at this stage will be equal to (X 1, X 3). The number 80 = 16 + 64. Therefore, the critical path at the second stage passes through work (X 3, X 4), etc. It is highlighted in the graph with a bold line:

X 1 - X 3 - X 4 - X 7 - X 8 - X 10 - X 11 - X 12 - X 13 - X 15 - X 16.

Early and late timing of events. Event time reserve

All paths that differ in duration from the critical one have time reserves. The difference between the length of the critical path and any non-critical path is called the total slack time of this non-critical path and is denoted by: .

Early term completion of an event is called the earliest point in time by which all work preceding this event is completed, i.e. is determined by the duration of the maximum path preceding the event, i.e.:

To find the early date of event j, you need to know the critical path of a directed subgraph consisting of the set of paths preceding this event j. The early date of the initial event is zero: t p (1)=0.

Late completion of an event is the latest point in time after which there remains exactly as much time as is necessary to complete all work following this event. The latest acceptable time for the event to occur, combined with the duration of all subsequent work, must not exceed the length of the critical path. The late date of the event is calculated as the difference between the duration of the critical path and the duration of the maximum path following the event:

For events lying on the critical path, the early and late dates for the completion of these events coincide.

The difference between the late and early dates of the event constitutes the event reserve time: . The interval is called the freedom interval of the event. The slack time of an event shows the maximum allowable time by which the moment of its occurrence can be postponed without increasing the critical path.

Since the amount determines the duration of the path of maximum length passing through this event, then, i.e. The time reserve of any event is equal to the full time reserve of the maximum path passing through this event.

When calculating time parameters manually, it is convenient to use the four-sector method. With this method, the circle of the network diagram indicating the event is divided into four sectors. The upper sector contains the event number; in the left - the earliest possible time of the event (); in the right - the latest permissible time of the event; in the lower sector - time reserve of this event: .

To calculate the early date of events: , apply the formula , considering events in ascending order of numbers, from initial to final, according to the works included in this event.

The late date of events is calculated using the formula , starting from the final event, for which ( is the number of the final event), according to the works coming out of it.

Critical events have a slack of zero. They define critical activities and the critical path.

Example 10.2. Let the network diagram shown in Fig. be given. 10.8.

Solution. Let's calculate the early dates of events:

So, the final event can occur only on the 14th day from the start of the project. This is the maximum time in which all project work can be completed. It is determined by the longest path. The early date of completion of work 6 =14 coincides with the critical time kr - the total duration of work lying on the critical path. Now you can highlight the work that belongs to the critical path, returning from the ending event to the starting event. Of the two jobs included in event 6, the length of the critical path determined jobs (5, 6), since (5 + 56)=14. Therefore, work (5, 6) is critical, etc. Works (1, 3), (3, 4), (4, 5), (5, 6) determined the critical path: kr = (1-3-4-5-6).

Let us now calculate the later dates of events. Let's put it. Let's use the dynamic programming method. All calculations will be carried out from the final event to the initial event. The latest dates for the events to occur are:

Since after event 5, to complete the project, you need to complete work (5, 6) lasting 3 days. There are two jobs coming out of event 4, so:

The slack time for event 2 is: . The reserves for the remaining events are zero, since these events are critical.

Early and late start and finish dates for work. Determination of work time reserves. Full reserve of work time.

The event immediately preceding this work will be called initial and denote , and the event immediately following it is final and designate . Then we will denote any work by . Knowing the timing of events, it is possible to determine the time parameters of the work.

Early start date equal to the early date of the event: .

Early completion date is equal to the sum of the early period of completion of the initial event and the duration of this work: or .

Late work completion date coincides with the late date of completion of its final event: .

Late start date is equal to the difference between the late date of completion of its final event and the amount of this work:

Since the deadlines for completing the work are within the boundaries determined by and, they can have different types of time reserves.

Full operating time reserve - it is the maximum time required to complete any job without exceeding the critical path. It is calculated as the difference between the late deadline for completing the final event and the early deadline for completing the work itself: . Since, then.

Thus, full operating time reserve is the maximum time by which its duration can be increased without changing the duration of the critical path. All non-critical work has a full slack time other than zero.

Free reserve work time- this is the amount of time that can be available when performing this work, provided that its initial and final events occur at their earliest dates: .

The network diagram is calculated in a tabular manner using the formulas previously set out in Section 4 (1-10). When determining the parameters of network models analytically, the calculation is performed in the form of a table. Let us consider the features of calculating network models using this method (application 1) using the example of calculating the parameters of the network diagram depicted in the assignment for this course work (option 15).

At the initial stage, it is necessary to describe the initial network model. In this case, the codes of all jobs and dependencies are entered in the first column of the table, starting with the job coming out of the first event. Job codes must be included in the table sequentially; arbitrary order of inclusion of jobs and dependencies in the table is unacceptable. The second column of the table contains the durations of all activities and dependencies.

The calculation of the network diagram begins with determining the values of the early work parameters. The early start of work 1-2 is equal to zero (formula 1), and its early end according to formula 2.

The early start of jobs 2-6 and 2-7 (in accordance with formula 3) is equal to the early finish of jobs 1-2.

The maximum early finish value of job 19-21, equal to 36, determines the duration of the critical path and, therefore, the total duration of execution of all jobs in the original network model. The resulting value of the early completion of this work 19-21 = 36 is transferred to the late completion column of the final work 20-21.

Late start of work 20-21 is determined in accordance with formula 5 (= 34)

The late start of work 20-21 is the late finish of the preceding work 15-20 (=).

Further, the calculation of later parameters is performed in the same way, except for cases when the job has several subsequent jobs (for example, job 6-9 has two subsequent ones - 9-10 and 9-14). In this case, in accordance with formula 4, the late finish of work 6-9 is equal to the minimum value of the late start of subsequent works 9-10 and 9-14.

To find the position of the critical path, it is necessary to determine the values of the total and private slack time for each job and dependency of the network diagram and enter their values, respectively, into columns 7 and 8 of the calculation table.

The total work time reserve, according to formulas 8-9, is defined as the difference between the late and early finishes or as the difference between the late and early starts of the corresponding work. It is useful to determine the value of the total slack using both methods; the coincidence of the obtained values can be considered as an additional check. For example, for work 6-7:

The partial work time reserve, according to formula 10, is defined as the difference between the early start value of the subsequent work and the early finish value for this work. For example, for work 6-7:

The critical path is characterized by zero slack time. A comparison of the network model parameters obtained by sector and tabular methods should reveal their complete identity; the presence of discrepancies indicates that the calculations are erroneous.

Graphical method for calculating network diagrams

Calculation of a network diagram graphically is carried out similarly to the tabular method (formulas 1-10), however, the graphical or sector method of calculating network diagram parameters involves recording them directly on the model (Appendix 2). In this case, each event (circle) is divided into four sectors. The designation of the sectors is shown in the following figure:

For activities on the critical path, the values of the total and private float are equal to zero; it is highlighted on the network diagram by a double line.

To check the correctness of the calculations performed, you should make sure that:

* a continuous critical path has been identified;
* calculated time reserves have a non-negative value;
* the value of the private time reserve for all jobs is less than or equal to the value of the general time reserve for these jobs;
* at least one late start value of jobs (jobs) coming from the first event is zero.

Two are known method for calculating network graph parameters." calculation directly on the network graph; analytical (tabular).

Calculation main indicators of the network model can produce as follows.

1. Calculation of early dates:
- ? early start of work determined by the duration of the longest path from the initial event to the start of this work,
- ? early completion dates- This is the earliest possible completion date for the work. The early start time of work is equal to the sum of the early start time of work and the duration of the work itself.
2.Calculation of the critical path. Its duration is defined as the total time of activities lying on the critical path, i.e. time for completion of the entire complex of work with the greatest parallelization of all work. This time is equal to the largest of the early completion times of the network graph shutdowns. The critical path passes through events that do not have time reserves (through critical activities).
3.Calculation of late start and finish dates for work are determined from the possibilities of a limiting shift to the right along the numerical axis of the work deadlines so that the critical path time is not changed. Therefore, it is logical to carry out calculations from the last event to the first and first determine the time of late completion of work, and then calculate the time of late start of work:
- ?late start date (ij) is defined as the difference between the late completion date of the work and the duration of the work itself,
- ? late completion date is determined by the value of the minimum duration path leading to it from the final event, and is calculated as the difference between the critical path and the maximum duration of the work from the final event of the network schedule to the final event of this work.
4. Calculation of time reserves."

Ifull operating time reserve defined as the difference between a late start and an early start or between a late finish and an early finish. It should be noted that the total time reserves for activities lying on the critical path are equal to zero,

? private (free) time reserves."
1)private time reserve of the first type determined by the ability to change the late start of work ( ij) to earlier dates without changing the later completion dates of immediately preceding work,
2) private reserve time of the second type determined by the ability to change the early end of work (ij) at a later date without changing the early dates for the start of immediately subsequent work; is determined by the difference between the early start of subsequent work and the early finish of this work.

Let's look at the procedure for calculating parameters using an example. The network diagram is shown in Fig. 7.5.

Rice. 7.5.

To calculate the parameters, we will use the tabular method, and in order to simplify perception, we will summarize everything in one table. 7.1.

Rules for the use of time reserves in network planning.

1. In order for the total and partial work reserves (y) to be equal, it is necessary and sufficient that the final event Y of the work in question is an event on the critical path.
2. If full reserve (Me and]1) some work is zero, then the private reserve of the second type (g"f) is also zero. There is always a relationship between these reserves R(IJ) > r" ijy Total and partial time reserves are always greater than or equal to zero.
3. In order for the partial reserve of work time (y) to be equal to zero, it is necessary and sufficient that this work lies on the path of maximum length from the first event to event y.
4. If the duration of work (y) is increased by the amount p, i.e. p then the early start date of subsequent work will increase by the amount p - g" (" yy
5. If the duration of work (y) is increased by the amount of the total reserve time of this work, then a new critical path is formed, the duration of which is equal to the duration of the old one.
6. The total reserve of work time (y) is equal to the sum of the private reserve of time of the second type of this work and the minimum of the total reserves of all immediately subsequent work.

Results of calculating network diagram parameters

Table 7.1

Duration	Early	terms, hours	Late dates, h		Time reserves, h
work, h	Beginnings	Endings	Beginnings	Endings	Full	Available












Critical path, h		(works 1-3

7. If the duration of work (g/) is increased by an amount p, then a new critical path will appear, the duration of which will exceed the duration of the old critical path by an amount p -

After the network diagram has been constructed and its main indicators have been calculated, we begin to optimize it.

1. Select the critical path and find its length;
2. Determine the time reserves for each event;
3. Determine the time reserves of all jobs and the work intensity coefficient of the penultimate job

Solution

To solve the problem, we use the following notation.

Network element	Parameter name	Parameter symbol
Event i	Early date of the event
Late event completion date
Event time reserve
Work (i, j)	Duration of work
Early start date
Early completion date
Late start date
Late work completion date
Full operating time reserve
	Travel duration
Critical path duration
Travel time reserve

To determine the time reserves for network events, the earliest t p and the latest t p dates of events are calculated. Any event cannot occur before all the events preceding it have occurred and all previous work has not been completed. Therefore, the early (or expected) time tp(i) of the i-th event is determined by the duration of the maximum path preceding this event:

t p (i) = max(t(L ni)) (1)

where L ni is any path preceding the i-th event, that is, the path from the initial to the i-th network event.

If event j has several previous paths, and therefore several previous events i, then the early date of event j is conveniently found using the formula:

t p (j) = max (2)

The delay in the completion of event i in relation to its earlier date will not affect the completion date of the final event (and therefore the completion period of the work package) until the sum of the completion period of this event and the duration (length) of the maximum of the following paths does not exceeds the length of the critical path. Therefore, the late (or deadline) date t p (i) for the completion of the i-th event is equal to:

t p (i) = t kp - max(t(L ci)) (3)

where Lci is any path following the i-th event, i.e. path from the i-th to the final network event.

If event i has several subsequent paths, and therefore several subsequent events j, then the late date for the completion of event i is conveniently found using the formula:

t p (i) = min

The time reserve R(i) of the i-th event is defined as the difference between the late and early dates of its occurrence:

R(i) = t p (i) - t p (i)

The reserve time of an event shows by what acceptable period of time the occurrence of this event can be delayed without causing an increase in the period of completion of the work package.

Critical events do not have time reserves, since any delay in the completion of an event lying on the critical path will cause the same delay in the completion of the final event. Thus, by determining the early date of the final event of the network, we thereby determine the length of the critical path.

When determining the early dates of events tp(i), we move along the network diagram from left to right and use formulas (1), (2).

Calculation of timing of events.

For i=0 (initial event), obviously tp(0)=0.

i=1: t p (1) = t p (0) + t(0,1) = 0 + 0 = 0.

i=2: t p (2) = t p (1) + t(1,2) = 0 + 8 = 8.

i=3: t p (3) = t p (1) + t(1,3) = 0 + 3 = 3.

i=4: max(t p (2) + t(2,4);t p (3) + t(3,4)) = max(8 + 6;3 + 3) = 14.

i=5: tp(5) = tp(4) + t(4,5) = 14 + 0 = 14.

i=6: max(t p (4) + t(4,6);t p (5) + t(5,6)) = max(14 + 5;14 + 3) = 19.

i=7: t p (7) = t p (6) + t (6,7) = 19 + 9 = 28.

i=8: max(t p (2) + t(2.8);t p (6) + t(6.8);t p (7) + t(7.8)) = max(8 + 18;19 + 5;28 + 4) = 32.

i=9: max(t p (5) + t(5,9);t p (7) + t(7,9)) = max(14 + 2;28 + 4) = 32.

i=10: max(t p (4) + t(4.10);t p (7) + t(7.10);t p (9) + t(9.10)) = max(14 + 4;28 + 2;32 + 0) = 32.

i=11: max(t p (8) + t(8,11);t p (10) + t(10,11)) = max(32 + 12;32 + 4) = 44.

The length of the critical path is equal to the early date of completion of the final event 11: t kp =tp(11)=44

When determining the late dates of events t p (i), we move through the network in the opposite direction, that is, from right to left and use formulas (3), (4).

For i=11 (final event), the late date of the event must be equal to its early date (otherwise the length of the critical path will change): t p (11)= t p (11)=44

i=10: t p (10) = t p (11) - t (10,11) = 44 - 4 = 40.

i=9: t p (9) = t p (10) - t (9,10) = 40 - 0 = 40.

All lines starting with number 8 are looked through.

i=8: t p (8) = t p (11) - t (8,11) = 44 - 12 = 32.

All lines starting with number 7 are looked through.

i=7: min(t p (8) - t(7.8);t p (9) - t(7.9);t p (10) - t(7.10)) = min(32 - 4;40 - 4;40 - 2) = 28.

i=6: min(t p (7) - t(6.7);t p (8) - t(6.8)) = min(28 - 9;32 - 5) = 19.

All lines starting with number 5 are looked through.

i=5: min(t p (6) - t(5.6);t p (9) - t(5.9)) = min(19 - 3;40 - 2) = 16.

i=4: min(t p (5) - t(4.5);t p (6) - t(4.6);t p (10) - t(4.10)) = min(16 - 0;19 - 5;40 - 4) = 14.

All lines starting with number 3 are looked through.

i=3: t p (3) = t p (4) - t (3,4) = 14 - 3 = 11.

i=2: min(t p (4) - t(2.4);t p (8) - t(2.8)) = min(14 - 6;32 - 18) = 8.

i=1: min(t p (2) - t(1,2);t p (3) - t(1,3)) = min(8 - 8;11 - 3) = 0.

(0,1): 0 - 0 = 0;

Table 1 - Event reserve calculation

Event number	Event timing: early tp(i)	Timing of the event: late tп(i)	Time reserve, R(i)

Filling out table 2.

We will move the list of works and their duration to the second and third columns. In this case, the work should be written down in column 2 sequentially: first starting from number 0, then from number 1, etc.

In the second column we will put a number characterizing the number of immediately preceding works (CPR) to the event from which the work in question begins.

So, for work (1,2) in column 1 we put the number 1, because number 1 ends with 1 jobs: (0,1).

Column 4 is obtained from Table 1 (t p (i)). Column 7 is obtained from Table 1 (t p (i)).

The values in column 5 are obtained by summing columns 3 and 4.

In column 6, the late start of work is defined as the difference between the late completion of these works and their duration (the data in column 3 is subtracted from the values of column 7);

The contents of column 8 (full reserve time R(ij)) is equal to the difference between columns 6 and 4 or columns 7 and 5. If R(ij) is zero, then the work is critical

Table 2 - Analysis of the network model over time

Work (i,j)	Number of previous works	Duration tij	Early dates: beginning tijР.Н.	Early dates: end of tijР.О.	Late dates: beginning tijP.N.	Late dates: end of tijP.O.	Time reserves: full RijП	Independent time reserve RijН	Private reserve of the first kind, Rij1	Private reserve type II, RijC

It should be noted that in addition to the full operating time reserve, there are three more types of reserves. Partial time reserve of the first type R 1 is part of the total time reserve by which the duration of the work can be increased without changing the late date of its initial event. R 1 is found by the formula:

R(i,j)= R p (i,j) - R(i)

The private time reserve of the second type, or the free time reserve Rc of work (i, j), is a part of the total time reserve by which the duration of the work can be increased without changing the early date of its final event. Rc is found by the formula:

R(i,j)= R p (i,j) - R(j)

The value of the free operating time reserve indicates the location of the reserves required for optimization.

Independent time reserve Rн work (i, j) - part of the total reserve obtained for the case when all previous jobs finish at a late date, and all subsequent ones begin at an early time. Rн is found by the formula:

R(i,j)= Rп(i,j)- R(i) - R(j)

Critical path: (0,1)(1,2)(2,4)(4,6)(6,7)(7,8)(8,11)

Critical path duration: 44

Let's find the work intensity coefficient of the penultimate job. Since the length of the critical path is 44, the maximum path passing through job (1.10) is 32, then

K(1.10)=(32-28)/(44-28)=0.296.

4. An Internet provider in a small city has 5 dedicated service channels. On average, it takes 25 minutes to serve one client. On average, the system receives 6 akzas per hour. If there are no free channels, a refusal follows. Determine service characteristics: probability of failure, average number of communication lines occupied by service, absolute and relative throughput, probability of service. Find the number of dedicated channels at which the relative system throughput will be at least 0.95. Assume that the flows of requests and services are the simplest

Service flow intensity:

Load intensity:

s = l * t obs = 6 * 25/60 = 2.5

Load intensity c = 2.5 shows the degree of consistency of the input and output flows of requests of the service channel and determines the stability of the queuing system.

The probability that the service:

1 channel busy:

p 1 = c 1 /1! p 0 = 2.5 1 /1! * 0.0857 = 0.214

2 channels are busy:

p 2 = c 2 /2! p 0 = 2.5 2 /2! * 0.0857 = 0.268

3 channels are busy:

p 3 = c 3 /3! p 0 = 2.5 3 /3! * 0.0857 = 0.223

4 channels are occupied:

p 4 = c 4 /4! p 0 = 2.5 4 /4! * 0.0857 = 0.139

Channel 5 busy:

p 5 = c 5 /5! p 0 = 2.5 5 /5! * 0.0857 = 0.0697

The probability of failure is the fraction rejected applications: