Mathematical models of the simplest queuing systems. The functions p0(t) and p1(t) determine the transition process in a single-channel QS and describe the process of exponential approach of the QS to its limit state with a characteristic time constant d
October 23, 2013 at 2:22 pmSqueak: Modeling Queuing Systems
- Programming,
- OOP,
- Parallel programming
There is very little information on Habré about such a programming language as Squeak. I will try to talk about it in the context of modeling queuing systems. I’ll show you how to write a simple class, explain its structure, and use it in a program that will serve requests through several channels.
A few words about Squeak
Squeak is an open, cross-platform implementation of the Smalltalk-80 programming language with dynamic typing and garbage collection. The interface is quite specific, but quite convenient for debugging and analysis. Squeak fully meets the concept of OOP. Everything is made up of objects, even structures if-then-else, for, while implemented with their help. The entire syntax boils down to sending a message to the object in the form:<объект> <сообщение>Any method always returns an object and a new message can be sent to it.
Squeak is often used for process modeling, but can also be used as a tool for creating multimedia applications and a variety of educational platforms.
Queuing systems
Queuing systems (QS) contain one or more channels that process requests coming from several sources. The time for servicing each request can be fixed or arbitrary, as well as the intervals between their receipt. This could be a telephone exchange, a laundry, cashiers in a store, a typing bureau, etc. It looks something like this:
The QS includes several sources that enter a common queue and are sent for servicing as processing channels become free. Depending on the specific features of real systems, the model may contain a different number of application sources and service channels and have different restrictions on the queue length and the associated possibility of loss of applications (refusals).
When modeling a QS, the problems of estimating the average and maximum queue length, the frequency of service failures, the average load of channels, and determining their number are usually solved. Depending on the task, the model includes software blocks for collecting, accumulating and processing the necessary statistical data on the behavior of processes. The most commonly used event flow models in QS analysis are regular and Poisson. Regular ones are characterized by the same time between the occurrences of events, and Poisson ones are characterized by random ones.
A little math
For a Poisson flow, the number of events X, falling within the length interval τ (tau), adjacent to the point t, distributed according to Poisson's law:Where a (t, τ)- average number of events occurring during a time interval τ .
The average number of events occurring per unit of time is λ(t). Therefore, the average number of events over a time interval τ , adjacent to the moment of time t, will be equal to:
Time T between two events λ(t) = const = λ distributed according to the law:
Distribution density of a random variable T has the form:
To obtain pseudo-random Poisson sequences of time intervals t i solve the equation:
Where r i- a random number uniformly distributed over the interval.
In our case this gives the expression:
Entire volumes could be written on random number generation. Here, to generate integers uniformly distributed over the interval, we use the following algorithm:
Where R i- another random integer;
R- some large prime number (for example 2311);
Q- integer - the upper limit of the interval, for example, 2 21 = 2097152;
rem- the operation of obtaining the remainder from division of integers.
Initial value R0 usually set arbitrarily, for example, using timer readings:
Time totalSeconds
To obtain numbers uniformly distributed over an interval, we use the language operator:
Rand class
To obtain random numbers uniformly distributed over an interval, we create a class - a real number generator:Float variableWordSubclass: #Rand "class name" instanceVariableNames: "" "instance variables" classVariableNames: "R" "class variables" poolDictionaries: "" "general dictionaries" category: "Sample" "category name"
Methods:
"Initialization" init R:= Time totalSeconds.next "Next pseudo-random number" next R:= (R * 2311 + 1) rem: 2097152. ^(R/2097152) asFloat
To set the initial state of the sensor, send a message Rand init.
To receive the next random number we send rand next.
Application Processing Program
So, as a simple example, let's do the following. Suppose we need to model servicing a regular flow of requests from one source with a random time interval between requests. There are two channels of different performance, allowing you to service requests in 2 and 7 units of time, respectively. It is necessary to register the number of requests served by each channel over an interval of 100 time units.Code for Squeak
"Declaring temporary variables" | proc1 proc2 t1 t2 s1 s2 sysPriority queue continue r | "Initial variable settings" Rand init. SysTime:= 0. s1:= 0. s2:= 0. t1:= -1. t2:= -1. continue:= true. sysPriority:= Processor activeProcess priority. "Current priority" queue:= Semaphore new. "Request queue model" "Creating a process - channel model 1" (Process forContext: [ proc1:= Processor activeProcess. whileTrue: "Service cycle" [ queue wait. "Wait for request" t1:= SysTime + 2. "Next activation time" s1:= s1 + 1. proc1 suspend. "Suspend the process while waiting for the end of service" ] proc1:= nil. "Remove the reference to process 1" ] priority: (sysPriority + 1)) resume. "The new priority is greater than the background" "Creating a process - channel model 2" (Process forContext: [ proc2:= Processor activeProcess.. whileTrue: [ queue wait. t2:= SysTime + 7. s2:= s2 + 1. proc2 suspend. ] .proc2:= nil. ] priority: (sysPriority + 1)) resume. "Continuation of the description of the main process and source model" whileTrue: [ r:= (Rand next * 10) rounded. (r = 0) ifTrue: . ((SysTime rem: r) = 0) ifTrue: . "Send request" "Service process switch" (t1 = SysTime) ifTrue: . (t2 = SysTime) ifTrue: . SysTime:= SysTime + 1. "Model time is ticking" ]. "Show order counter status" PopUpMenu inform: "proc1: ",(s1 printString),", proc2: ",(s2 printString). continue:= false.
When launched, we see that process 1 managed to process 31 requests, and process 2 only 11:
Classification, basic concepts, model elements, calculation of basic characteristics.
When solving problems of rational organization of trade, consumer services, warehousing, etc. It is very useful to interpret the activities of the production structure as queuing systems, i.e. a system in which, on the one hand, there are constantly requests to perform some work, and on the other hand, these requests are constantly satisfied.
Every QS includes four elements: incoming stream, queue, server, outgoing stream.
Requirement(client, application) in the QS is each individual request to perform any work.
Service- this is the performance of work to satisfy the received requirement. The object that performs servicing of requests is called a service device (device) or service channel.
Service time is the period during which a service request is satisfied, i.e. the period from the start of service to its completion. The period from the moment a request enters the system until the start of service is called the service waiting time. The waiting time for service, together with the service time, constitutes the time the request remains in the system.
SMOs are classified according to different criteria.
1. Based on the number of service channels, QSs are divided into single-channel and multi-channel.
2. Depending on the waiting conditions and the requirement to start servicing, a distinction is made between QS with losses (failures) and QS with waiting.
IN QS with loss requirements, received at a time when all devices are busy with servicing, are rejected, they are lost for this system and do not have any effect on the further servicing process. A classic example of a faulty system is the telephone exchange - a connection request is refused if the called subscriber is busy.
For a system with failures, the main characteristic of operational efficiency is the probability of failure or the average proportion of applications remaining unserved.
IN QS with expectation of requirement, arriving at a time when all devices are busy servicing, does not leave the system, but gets into a queue and waits until one of the channels is free. When the next device becomes available, one of the queued requests is immediately accepted for service.
For a QS with waiting, the main characteristics are the mathematical expectations of the queue length and waiting time.
An example of a waiting system is the process of restoring televisions in a repair shop.
There are systems lying between these two groups ( mixed SMO). They are characterized by the presence of some intermediate conditions: restrictions may be restrictions on the waiting time for the start of service, on the length of the queue, etc.
Performance characteristics can be applied to the probability of failure in both lossy systems (or latency characteristics) and waiting systems.
3. According to the maintenance discipline, QS systems are divided into systems with priority in maintenance and systems without priority in maintenance.
Requests can be serviced in the order they are received, either randomly or based on established priorities.
4. SMO can be single-phase and multi-phase.
IN single-phase systems, requirements are served by channels of one type (for example, workers of the same profession) without transferring them from one channel to another, in multiphase systems such transfers are possible.
5. Based on the location of the source of requirements, QS systems are divided into open (when the source of the requirement is outside the system) and closed (when the source is in the system itself).
TO closed These include systems in which the incoming flow of demands is limited. For example, a foreman whose task is to set up machines in a workshop must periodically service them. Each adjusted machine becomes a potential source of adjustment requirements in the future. In such systems, the total number of circulating requirements is finite and most often constant.
If the supply source has an infinite number of requirements, then the systems are called open. Examples of such systems include shops, ticket offices at train stations, ports, etc. For these systems, the incoming flow of demands can be considered unlimited.
Methods and models for researching QS can be divided into analytical and statistical (simulation modeling of queuing processes).
Analytical methods make it possible to obtain the characteristics of a system as some functions of the parameters of its functioning. Thanks to this, it becomes possible to conduct a qualitative analysis of the influence of individual factors on the efficiency of the QS.
Unfortunately, only a fairly limited range of problems in queuing theory can be solved analytically. Despite the ongoing development of analytical methods, in many real cases an analytical solution is either impossible to obtain, or the resulting dependencies turn out to be so complex that their analysis becomes a difficult task in itself. Therefore, in order to be able to use analytical solution methods, one has to resort to various simplifying assumptions, which is to some extent compensated by the possibility of using a qualitative analysis of the final dependencies (in this case, of course, it is necessary that the assumptions made do not distort the real picture of the process).
Currently, the most theoretically developed and convenient in practical applications are methods for solving queuing problems in which the flow of requirements is the simplest ( Poisson).
For the simplest flow, the frequency of arrival of requirements into the system obeys Poisson’s law, that is, the probability of arrival in time t equal to k requirements is given by the formula:
where λ is the flow parameter (see below).
The simplest flow has three main properties: ordinary, stationary and lack of aftereffect.
Ordinariness flow means the practical impossibility of simultaneous arrival of two or more demands. For example, the probability is quite small that from a group of machines serviced by a team of repairmen, several machines will fail at the same time.
Stationary called flow, for which the mathematical expectation of the number of demands entering the system per unit time (denoted by λ) does not change over time. Thus, the probability of a certain number of demands entering the system during a given period of time Δt depends on its value and does not depend on the beginning of its counting on the time axis.
No aftereffect means that the number of demands received into the system before time t does not determine how many demands will enter the system during time t + Δt.
For example, if a thread breakage occurs on a weaving loom at a given moment, and it is repaired by the weaver, then this does not determine whether a new breakage will occur on this loom at the next moment or not, much less does it affect the likelihood of a breakage occurring on other looms.
An important characteristic of a QS is the time it takes to service requirements in the system. Service time is, as a rule, a random variable and, therefore, can be described by a distribution law. The most widely used in theory and, especially in practical applications, is the exponential law. For this law, the probability distribution function has the form:
F(t) = 1 – e -μt ,
those. the probability that the service time does not exceed a certain value t is determined by the formula (1 – e -μt), where μ is the parameter of the exponential law of service time for requirements in the system - the reciprocal of the average service time, i.e. .
Let's consider analytical models of QS with expectation(the most common QS, in which requests received when all service units are busy are queued and serviced as service units are released).
Tasks with queues are typical in production environments, for example, when organizing adjustment and repair work, during multi-machine maintenance, etc.
The general formulation of the problem is as follows.
The system consists of n serving channels. Each of them can serve only one requirement at a time. The system receives a simple (Poisson) flow of demands with parameter λ. If, at the time the next request arrives, there are already at least n requests in the system for servicing (i.e., all channels are busy), then this request becomes queued and waits for servicing to begin.
The service time of each request t about is a random variable that obeys an exponential distribution law with the parameter μ.
As noted above, QS with expectation can be divided into two large groups: closed and open.
The peculiarities of the functioning of each of these two types of systems impose their own shade on the mathematical apparatus used. Calculation of the operating characteristics of various types of QS can be carried out on the basis of calculating the probabilities of QS states (Erlang formulas).
Since the system is closed, a condition should be added to the problem statement: the flow of incoming requirements is limited, i.e. there cannot be more than m requirements in the service system at the same time (m is the number of objects being serviced).
As the main criteria characterizing the quality of functioning of the system under consideration, we will choose: 1) the ratio of the average queue length to the largest number of demands simultaneously located in the serving system - the downtime rate of the serviced object; 2) the ratio of the average number of idle serving channels to their total number - the idle ratio of the serviced channel.
Let's consider the calculation of the necessary probabilistic characteristics (performance indicators) of a closed QS.
1. The probability that there are k requirements in the system, provided that their number does not exceed the number of service devices n:
P k = α k P 0 , (1 ≤ k ≤ n),
Where 
λ is the frequency (intensity) of requirements entering the system from one source;
Average duration of servicing one request;
m is the largest possible number of requirements located in the serving system at the same time;
n - number of service devices;
P 0 is the probability that all service devices are free.
2. The probability that there are k demands in the system, provided that their number is greater than the number of service devices:
P k = α k P 0 , (n ≤ k ≤ m),
Where 
3. The probability that all service devices are free is determined from the condition
hence,
4. Average number of requests waiting for service to begin (average queue length):
5. Request downtime rate awaiting service:
6. Probability that all service devices are busy:
7. Average number of requirements in the serving system (served and awaiting service):
8. Rate of complete downtime of requirements for maintenance and awaiting maintenance:
9. Average downtime of a request in the queue for service:
10. Average number of free service devices:
11. Downtime ratio of service devices:
12. The probability that the number of demands waiting for service is greater than a certain number B (the probability that there are more than B demands in the queue for service):
In many areas of economics, finance, production and everyday life, systems that implement repeated execution of similar tasks play an important role. Such systems are called queuing systems ( CMO ). Examples of QS are: banks of various types, insurance organizations, tax inspectorates, audit services, various communication systems, loading and unloading complexes, gas stations, various enterprises and service organizations.
3.1.1 General information about queuing systems
Each QS is designed to service (fulfill) a certain flow of applications (requirements) that arrive at the input of the system, mostly not regularly, but at random times. Service of applications also does not last a constant, predetermined time, but a random one, which depends on many random, sometimes unknown to us, reasons. After servicing the request, the channel is freed and ready to receive the next request. The random nature of the flow of applications and the time of their servicing leads to uneven workload of the QS. In some time intervals, requests may accumulate at the input of the QS, which leads to an overload of the QS; in some other time intervals, when there are free channels (service devices) at the input of the QS, there will be no requests, which leads to an underload of the QS, i.e. to idleness of its channels. Applications that accumulate at the entrance of the QS either “join” the queue, or for some reason they cannot continue to stay in the queue, they leave the QS unserved.
Figure 3.1 shows the QS diagram.
The main elements (features) of queuing systems are:
Service unit (block),
Flow of applications
Queue while waiting for service (queuing discipline).
Service block designed to perform actions in accordance with the requirements of incoming system applications.
Rice. 3.1 Queuing system diagram
The second component of queuing systems is the input flow of applications. Applications are entered into the system randomly. It is usually assumed that the input stream obeys a certain probability law for the duration of the intervals between two successively arriving requests, and the distribution law is considered to be unchanged for some sufficiently long time. The source of applications is unlimited.
The third component is queue discipline. This characteristic describes the order of service of requests arriving at the input of the system. Since the serving block usually has a limited capacity, and requests arrive irregularly, a queue of requests is periodically created waiting for service, and sometimes the serving system is idle waiting for requests.
The main feature of queuing processes is randomness. In this case, there are two interacting parties: served and serving. Random behavior of at least one of the parties leads to the random nature of the flow of the service process as a whole. The sources of randomness in the interaction of these two parties are random events of two types.
1. Appearance of an application (requirement) for service. The reason for the randomness of this event is often the massive nature of the need for service.
2. End of servicing of the next application. The reasons for the randomness of this event are both the randomness of the start of the service and the random duration of the service itself.
These random events constitute a system of two flows in the QS: the input flow of applications for service and the output flow of serviced applications.
The result of the interaction of these streams of random events is the number of applications currently in the QS, which is usually called the state of the system.
Each QS, depending on its parameters of the nature of the flow of applications, the number of service channels and their productivity, and the rules of work organization, has a certain operating efficiency (throughput) that allows it to successfully cope with the flow of applications.
Special field of applied mathematics mass theorymaintenance (TMO)– deals with process analysis in queuing systems. The subject of study of queuing theory is QS.
The goal of queuing theory is to develop recommendations for the rational construction of QS, rational organization of their work and regulation of the flow of requests to ensure high efficiency of QS operation. To achieve this goal, the tasks of queuing theory are set, which consist in establishing the dependencies of the effectiveness of the functioning of the QS on its organization.
The problems of queuing theory are of an optimization nature and are ultimately aimed at determining a version of the system that will ensure a minimum of total costs from waiting for service, loss of time and resources for service, and downtime of the service unit. Knowledge of such characteristics provides the manager with information to develop targeted influence on these characteristics to manage the efficiency of queuing processes.
The following three main groups of (usually average) indicators are usually selected as characteristics of the effectiveness of the QS system:
Indicators of effectiveness of using QS:
The absolute capacity of the QS is the average number of requests that the QS can serve per unit of time.
Relative capacity of the QS is the ratio of the average number of applications served by the QS per unit of time to the average number of applications received during the same time.
Average duration of the employment period of the CMO.
QS utilization rate is the average share of time during which the QS is busy servicing requests, etc.
Quality indicators for servicing requests:
Average waiting time for an application in the queue.
Average time an application stays in the CMO.
The probability of a request being denied service without waiting.
The probability that an received application will be immediately accepted for service.
The law of distribution of the time an application stays in the queue.
The law of distribution of the time an application stays in the QS.
The average number of applications in the queue.
Average number of applications in the CMO, etc.
Indicators of the effectiveness of the pair “SMO - consumer”, where “consumer” is understood as the entire set of applications or some of them
For QS with failures:
For SMO with unlimited waiting both absolute and relative throughput lose their meaning, since every incoming request will sooner or later be serviced. For such a QS, the important indicators are:
For Mixed type QS both groups of indicators are used: both relative and absolute throughput, and characteristics of expectation.
Depending on the purpose of the queuing operation, any of the given indicators (or a set of indicators) can be selected as an efficiency criterion.
Analytical model A QS is a set of equations or formulas that allow one to determine the probabilities of system states during its operation and calculate performance indicators based on the known characteristics of the incoming flow and service channels.
There is no general analytical model for an arbitrary QS. Analytical models have been developed for a limited number of special cases of QS. Analytical models that more or less accurately reflect real systems are usually complex and difficult to visualize.
Analytical modeling of a QS is greatly facilitated if the processes occurring in the QS are Markovian (the flows of requests are simple, service times are distributed exponentially). In this case, all processes in the QS can be described by ordinary differential equations, and in the limiting case, for stationary states, by linear algebraic equations and, having solved them, the selected efficiency indicators can be determined.
Let's look at examples of some QS.
2.5.1. Multichannel QS with failures
Example 2.5. Three traffic inspectors check the waybills of truck drivers. If at least one inspector is free, the passing truck is stopped. If all the inspectors are busy, the truck passes by without stopping. The flow of trucks is simple, the check time is random with an exponential distribution.
This situation can be modeled by a three-channel QS with failures (no queue). The system is open-loop, with homogeneous requests, single-phase, with absolutely reliable channels.
Description of states:
All inspectors are free;
One inspector is busy;
Two inspectors are busy;
Three inspectors are busy.
The system state graph is shown in Fig. 2.11.
Rice. 2.11.
On the graph: - intensity of truck flow; - intensity of document checks by one traffic inspector.
Simulation is carried out to determine the portion of vehicles that will not be tested.
Solution
The required part of the probability is the probability of employment of all three inspectors. Since the state graph represents a typical “death and reproduction” scheme, we will find using dependencies (2.2).
The throughput capacity of this traffic inspector post can be characterized relative throughput:
Example 2.6. To receive and process reports from the reconnaissance group, a group of three officers was appointed in the intelligence department of the association. The expected intensity of the flow of reports is 15 reports per hour. The average time for processing one report by one officer is . Each officer can receive reports from any reconnaissance group. The released officer processes the last of the received reports. Incoming reports must be processed with a probability of at least 95%.
Determine whether the assigned team of three officers is sufficient to complete the assigned task.
Solution
A group of officers operates as a CMO with failures, consisting of three channels.
Flow of reports with intensity 
can be considered the simplest, since it is the total of several reconnaissance groups. Service intensity 
. The distribution law is unknown, but this is unimportant, since it has been shown that for systems with failures it can be arbitrary.
The description of states and the state graph of the QS will be similar to those given in example 2.5.
Since the state graph is a “death and reproduction” scheme, there are ready-made expressions for it for the limiting probabilities of the state:
The attitude is called given intensity of the flow of applications. Its physical meaning is as follows: the value represents the average number of requests arriving at the QS during the average time of servicing one request.
In the example 
.
In the QS under consideration, a failure occurs when all three channels are busy, that is. Then:
Because probability of failure in the processing of reports is more than 34% (), then it is necessary to increase the personnel of the group. Let's double the composition of the group, that is, the CMO will now have six channels, and calculate:
Thus, only a group of six officers will be able to process incoming reports with a 95% probability.
2.5.2. Multi-channel QS with waiting
Example 2.7. At the river crossing section there are 15 similar crossing facilities. The flow of equipment arriving at the crossing is on average 1 unit/min, the average time for crossing one unit of equipment is 10 minutes (including the return of the crossing vehicle).
Assess the main characteristics of the crossing, including the likelihood of immediate crossing immediately upon the arrival of the unit of equipment.
Solution
Absolute throughput, i.e. everything that approaches the crossing is practically crossed immediately.
Average number of operating crossing facilities:
Ferry utilization and downtime rates:
A program was also developed to solve the example. The time intervals for equipment to arrive at the crossing and the crossing time are assumed to be distributed according to an exponential law.
The utilization rates of the crossing after 50 runs are almost the same: 
.
INTRODUCTION
CHAPTER I. FORMULATION OF QUEUE SERVICE PROBLEMS
1.1 General concept of queuing theory
1.2 Modeling of queuing systems
1.3 QS state graphs
1.4 Random processes
Chapter II. EQUATIONS DESCRIBING QUEUE SYSTEMS
2.1 Kolmogorov equations
2.2 Processes of “birth - death”
2.3 Economic and mathematical formulation of queuing problems
Chapter III. MODELS OF QUEUING SYSTEMS
3.1 Single-channel QS with denial of service
3.2 Multi-channel QS with denial of service
3.3 Model of a multiphase tourist service system
3.4 Single-channel QS with limited queue length
3.5 Single-channel QS with unlimited queue
3.6 Multi-channel QS with limited queue length
3.7 Multi-channel QS with unlimited queue
3.8 Analysis of the supermarket queuing system
CONCLUSION
Introduction
Currently, a large amount of literature has appeared devoted directly to the theory of queuing, the development of its mathematical aspects, as well as various areas of its application - military, medical, transport, trade, aviation, etc.
Queuing theory is based on probability theory and mathematical statistics. The initial development of queuing theory is associated with the name of the Danish scientist A.K. Erlang (1878-1929), with his works in the field of design and operation of telephone exchanges.
Queuing theory is a field of applied mathematics that deals with the analysis of processes in production, service, and management systems in which homogeneous events are repeated many times, for example, in consumer service enterprises; in systems for receiving, processing and transmitting information; automatic production lines, etc. A great contribution to the development of this theory was made by Russian mathematicians A.Ya. Khinchin, B.V. Gnedenko, A.N. Kolmogorov, E.S. Wentzel et al.
The subject of queuing theory is to establish dependencies between the nature of the flow of requests, the number of service channels, the performance of an individual channel and effective service in order to find the best ways to manage these processes. The problems of queuing theory are of an optimization nature and ultimately include the economic aspect of determining a system option that will ensure a minimum of total costs from waiting for service, loss of time and resources for service, and downtime of service channels.
In commercial activities, the application of queuing theory has not yet found the desired distribution.
This is mainly due to the difficulty of setting tasks, the need for a deep understanding of the content of commercial activities, as well as reliable and accurate tools that allow one to calculate various options for the consequences of management decisions in commercial activities.
Chapter I . Setting queuing tasks
1.1 General concept of queuing theory
The nature of mass services, in various fields, is very subtle and complex. Commercial activity is associated with the performance of many operations at the stages of movement, for example, the mass of goods from the sphere of production to the sphere of consumption. Such operations are loading of goods, transportation, unloading, storage, processing, packaging, and sales. In addition to such basic operations, the process of movement of goods is accompanied by a large number of preliminary, preparatory, accompanying, parallel and subsequent operations with payment documents, containers, money, cars, clients, etc.
The listed fragments of commercial activity are characterized by the mass receipt of goods, money, visitors at random times, then their consistent service (satisfaction of requirements, requests, requests) by performing appropriate operations, the execution time of which is also random. All this creates unevenness in work, generates underloads, downtime and overloads in commercial operations. A lot of trouble is caused by queues, for example, visitors in cafes, canteens, restaurants, or car drivers at commodity depots, waiting for unloading, loading or paperwork. In this regard, there are tasks of analyzing the existing options for performing the entire set of operations, for example, the trading floor of a supermarket, a restaurant, or in workshops for the production of own products in order to evaluate their work, identify weak links and reserves, and ultimately develop recommendations aimed at increasing the efficiency of commercial operations. activities.
In addition, other tasks arise related to the creation, organization and planning of a new economical, rational option for performing many operations within the trading floor, confectionery shop, all service levels of a restaurant, cafe, canteen, planning department, accounting department, personnel department, etc.
The tasks of organizing mass service arise in almost all spheres of human activity, for example, servicing buyers in stores by sellers, serving visitors at catering establishments, servicing customers at consumer services enterprises, providing telephone conversations at a telephone exchange, providing medical care to patients in a clinic, etc. . In all the above examples, there is a need to satisfy the needs of a large number of consumers.
The listed problems can be successfully solved using methods and models of queuing theory (QST) specially created for these purposes. This theory explains that it is necessary to serve someone or something, which is defined by the concept of “service request (demand)”, and service operations are performed by someone or something called service channels (nodes). The role of requests in commercial activities is played by goods, visitors, money, auditors, documents, and the role of service channels is played by sellers, administrators, cooks, confectioners, waiters, cashiers, merchandise experts, loaders, commercial equipment, etc. It is important to note that in one embodiment, for example, a cook in the process of preparing dishes is a service channel, and in another he acts as a request for service, for example to the production manager to receive goods.
Applications, due to the massive number of receipts for servicing, form flows that are called incoming before servicing operations are performed, and after a possible wait for the start of servicing, i.e. idle time in the queue form service flows in the channels, and then an outgoing flow of requests is formed. In general, the combination of elements of the incoming flow of requests, a queue, service channels and the outgoing flow of requests forms the simplest single-channel queuing system - QS.
A system is understood as a set of interconnected systems. purposefully interacting parts (elements). Examples of such simple QS in commercial activities are places for receiving and processing goods, payment centers for customers in stores, cafes, canteens, workplaces for economists, accountants, merchants, cooks, etc.
The service procedure is considered completed when the service request leaves the system. The duration of the time interval required to implement the service procedure depends mainly on the nature of the request for service, the state of the service system itself and the service channel.
Indeed, the length of a buyer’s stay in a supermarket depends, on the one hand, on the personal qualities of the buyer, his requests, on the range of goods that he is going to purchase, and on the other hand, on the form of service organization and service personnel, which can significantly affect the buyer’s stay. in the supermarket and the intensity of service. For example, mastering the “blind” method of working at a cash register by cashiers-controllers made it possible to increase the throughput of payment nodes by 1.3 times and save the time spent on settlements with customers at each cash register by more than 1.5 hours per day. The introduction of a single payment center in a supermarket provides tangible benefits to the buyer. Thus, if with the traditional form of payment the time for servicing one customer was on average 1.5 minutes, then with the introduction of a single payment unit it was 67 seconds. Of these, 44 seconds are spent on making a purchase in the section and 23 seconds directly on payments for purchases. If the buyer makes several purchases in different sections, then the loss of time is reduced when purchasing two purchases by 1.4 times, three by 1.9, five by 2.9 times.
By servicing requests we mean the process of satisfying a need. Service is different in nature. However, in all examples, received requests require servicing by some device. In some cases, the service is performed by one person (service to the buyer by one seller, in some - by a group of people (service to a patient by a medical commission in a clinic), and in some cases - by technical devices (sale of sparkling water, sandwiches by vending machines). A set of means that service requests , is called a service channel.
If service channels are capable of satisfying identical requests, then service channels are called homogeneous. A set of homogeneous service channels is called a service system.
The queuing system receives a large number of requests at random times, the service duration of which is also a random variable. The sequential arrival of applications into the service system is called the incoming flow of applications, and the sequence of applications leaving the service system is called the outgoing flow.
