Mathematical models of the simplest queuing systems. The functions p0(t) and p1(t) determine the transient process in a single-channel QS and describe the process of QS exponentially approaching its limit state with a characteristic time constant d
October 23, 2013 at 02:22 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 will show how to write a simple class, describe 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 a garbage collector. The interface is quite specific, but quite convenient for debugging and analysis. Squeak fully complies with the concept of OOP. Everything is made up of objects, even structures if-then-else, for, while implemented with their help. The whole 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 applications from several sources. The time for servicing each request can be fixed or arbitrary, as well as the intervals between their arrival. It can 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 the common queue and are sent for servicing as the processing channels become free. Depending on the specific features of real systems, the model may contain a different number of request sources and service channels and have different restrictions on the queue length and the associated possibility of losing requests (failures).
When modeling a QS, the tasks of estimating the average and maximum queue lengths, denial of service frequency, average channel load, 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 occurrence of events, while Poisson ones are random.
A bit of 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 the Poisson law:Where a (t, τ)- the average number of events occurring in the time interval τ .
The average number of events occurring per unit of time is equal to λ(t). Therefore, the average number of events per time interval τ , adjoining 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 looks like:
To obtain pseudo-random Poisson sequences of time intervals t i solve the equation:
Where r i is a random number uniformly distributed over the interval.
In our case, this gives the expression:
By generating random numbers, you can write entire volumes. 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 (eg 2311);
Q- integer - the upper limit of the interval, for example, 2 21 = 2097152;
rem- the operation of obtaining the remainder from the division of integers.
Initial value R0 usually set arbitrarily, for example, using the timer readings:
Time totalSeconds
To obtain numbers evenly distributed over the interval, we use the language operator:
Rand class
To obtain random numbers uniformly distributed over the interval, we create a class - a generator of real numbers:Float variableWordSubclass: #Rand "class name" instanceVariableNames: "" "instance variables" classVariableNames: "R" "class variables" poolDictionaries: "" "common 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 get another random number, send Rand next.
Application Processing Program
So, as a simple example, let's do the following. Suppose we need to simulate the maintenance of a regular flow of requests from one source with a random time interval between requests. There are two channels of different performance, which allow servicing applications in 2 and 7 units of time, respectively. It is necessary to register the number of requests served by each channel in the interval of 100 time units.Squeak Code
"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. "Claim Queue Model" "Creating Process - Channel Model 1" s1:= s1 + 1. proc1 suspend."Suspend process pending service termination" ].proc1:= nil."Remove reference to process 1" ]priority: (sysPriority + 1)) resume. "New priority is greater than background" "Create process - channel model 2" .proc2:= nil.] priority: (sysPriority + 1)) resume. "Continuing description of 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 request counter status" PopUpMenu inform: "proc1: ",(s1 printString),", proc2: ",(s2 printString). continue:= false.
At startup, we see that process 1 managed to process 31 requests, and process 2 only 11:
Classification, basic concepts, model elements, calculation of the main characteristics.
When solving problems of rational organization of trade, consumer services, warehousing, etc. very useful is the interpretation of the activities of the production structure as queuing systems, i.e. a system in which, on the one hand, requests for the performance of any work constantly arise, and on the other hand, these requests are constantly satisfied.
Every SMO includes four elements: incoming stream, queue, server, outgoing stream.
requirement(client, application) in the QS is each individual request for the performance of any work.
Service is the execution of work to satisfy the incoming demand. The object that performs the maintenance of requirements is called a service device (device) or a service channel.
The service time is the period during which the service requirement is satisfied, i.e. the period from the beginning of the service to its completion. The period from the moment a request enters the system to the start of service is called the service wait time. The waiting time for service, together with the service time, is the residence time of the requirement in the system.
SMOs are classified according to different criteria..
1. According to the number of service channels, QS are divided into single-channel and multi-channel.
2. Depending on the waiting conditions, the service start requirement distinguishes QS with losses (failures) and QS with waiting.
IN QS with loss of demand, received at the moment when all devices are busy with maintenance, are rejected, they are lost for this system and have no effect on the further maintenance process. The classic example of a failing system is the telephone exchange - a connection request is denied if the called party is busy.
For a system with failures, the main characteristic of the efficiency of functioning is the probability of failure or the average proportion of requests that remain unserved.
IN CMO with demand waiting, received at the moment when all devices are busy servicing, does not leave the system, but queues up and waits until one of the channels becomes free. When the next device is released, one of the applications in the queue is immediately accepted for service.
For QS with waiting, the main characteristics are the mathematical expectations of the queue length and waiting time.
An example of a wait-and-see system is the process of restoring televisions in a repair shop.
There are systems that lie between these two groups ( mixed CMOs). They are characterized by the presence of some intermediate conditions: restrictions can be restrictions on the waiting time for the start of service, on the length of the queue, etc.
As performance characteristics, the probability of failure can be used both in systems with losses (or characteristics of waiting time) and in systems with waiting.
3. According to the service discipline, QSs are divided into systems with service priority and systems without service priority.
Requests can be serviced in the order in which they are received, either randomly or based on established priorities.
4. QS can be single-phase and multi-phase.
IN single-phase systems, requirements are served by channels of the same type (for example, workers of the same profession) without transferring them from one channel to another, in multiphase systems such transfers are possible.
5. According to the location of the source of requirements, the QS 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 include systems in which the incoming flow of requirements is limited. For example, a foreman whose task is to set up machines in the workshop must periodically service them. Each set up machine becomes a potential source of set-up requirements in the future. In such systems, the total number of circulating claims 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 are shops, ticket offices of stations, ports, etc. For these systems, the incoming flow of requests can be considered unlimited.
Methods and models for studying QS can be conditionally divided into analytical and statistical (simulation modeling of queuing processes).
Analytical methods make it possible to obtain the characteristics of the system as some functions of the parameters of its functioning. This makes it possible to conduct a qualitative analysis of the influence of individual factors on the efficiency of the QS.
Unfortunately, only a rather 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 an independent difficult task. Therefore, in order to be able to apply analytical methods of solution, one has to resort to various simplifying assumptions, which is to some extent compensated by the possibility of applying 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).
At present, theoretically, the most developed and convenient in practical applications are methods for solving such queuing problems in which the flow of requirements is the simplest ( Poisson).
For the simplest flow, the frequency of receipt of requirements into the system obeys the Poisson 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 no aftereffect.
Ordinariness flow means the practical impossibility of simultaneous receipt of two or more requirements. For example, the probability that several machines from a group of machines serviced by a team of repairmen will fail at the same time is quite small.
Stationary called flow, for which the mathematical expectation of the number of claims entering the system per unit time (denoted by λ) does not change in time. Thus, the probability of a certain number of claims entering the system during a given time interval Δt depends on its value and does not depend on its origin on the time axis.
No aftereffect means that the number of customers entering the system before time t does not determine how many customers will enter the system in time t + Δt.
For example, if a thread break occurs on a loom at the moment, and it is eliminated by the weaver, then this does not determine whether a new break will occur on this loom at the next moment or not, all the more so it does not affect the probability of a break on other machines.
An important characteristic of QS is the service time of requirements in the system. The service time is, as a rule, a random variable and, therefore, can be described by a distribution law. The exponential law has received the greatest distribution in theory and, especially in practical applications. For this law, the probability distribution function has the form:
F(t) \u003d 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 the service time of requirements in the system - the reciprocal of the average service time, i.e. .
Consider analytical QS models with expectation(the most common QS, in which the requests received at the moment when all service units are busy are queued and serviced as the service units become free).
Tasks with queues are typical in production conditions, for example, when organizing adjustment and repair work, during multi-machine maintenance, etc.
The general problem statement is as follows.
The system consists of n serving channels. Each of them can serve only one request at a time. The system receives the simplest (Poisson) flow of requirements with the parameter λ. If at the moment of arrival of the next request in the system there are already at least n requests in service (i.e., all channels are busy), then this request enters the queue and waits for service to begin.
The service time of each requirement t about is a random variable that obeys the exponential distribution law with the parameter μ.
As noted above, QS with expectation can be divided into two large groups: closed and open.
The features of the functioning of each of these two types of systems impose their own shade on the mathematical apparatus used. The calculation of the characteristics of the QS operation of various types can be carried out on the basis of the calculation of 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 requests is limited, i.e. the queuing system cannot have more than m requests at the same time (m is the number of serviced objects).
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 requirements that are simultaneously in the servicing system - the downtime coefficient of the serviced object; 2) the ratio of the average number of idle serving channels to their total number is the idle ratio of the served channel.
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 receipt of requirements into the system from one source;
Average duration of service of one requirement;
m - the largest possible number of requirements that are in the serving system at the same time;
n is the number of service devices;
P 0 - the probability that all service devices are free.
2. The probability that there are k requirements 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 servers are free is determined from the condition
hence,
4. Average number of requests waiting to start service (average queue length):
5. Demand downtime ratio waiting for service:
6. The probability that all service devices are busy:
7. The average number of requirements in the service system (served and waiting for service):
8. The ratio of total downtime of requirements for service and waiting for service:
9. Average idle time of a claim in a service queue:
10. Average number of free attendants:
11. Downtime ratio of service vehicles:
12. The probability that the number of customers waiting to be serviced is greater than some number B (the probability that there are more than B customers in the service queue):
In many areas of the economy, finance, production and everyday life, systems that implement the repeated execution of tasks of the same type play an important role. Such systems are called queuing systems ( CMO ). Examples of SMOs are: banks of various types, insurance organizations, tax inspectorates, audit services, various communication systems, loading and unloading complexes, gas stations, various enterprises and organizations in the service sector.
3.1.1 General information about queuing systems
Each QS is designed to serve (execute) a certain flow of applications (requirements) that arrive at the input of the system for the most part not regularly, but at random times. The service of applications also lasts not for a constant, predetermined time, but random, which depends on many random, sometimes unknown to us, reasons. After servicing the request, the channel is released and is ready to receive the next request. The random nature of the flow of applications and the time of their service leads to an uneven workload of the QS. At some time intervals, requests can accumulate at the QS input, which leads to QS overload, while at some other time intervals, with free channels (service devices), there will be no requests at the QS input, which leads to QS underload, i.e. to idle its channels. Applications that accumulate at the entrance of the QS either “get” into the queue, or, for some reason, the impossibility of further staying in the queue, leave the QS unserved.
Figure 3.1 shows a diagram of the QS.
The main elements (features) of queuing systems are:
Service node (block),
application flow,
Queue waiting for service (queue discipline).
Service block designed to perform actions in accordance with the requirements of incoming system applications.
Rice. 3.1 Scheme of the queuing system
The second component of queuing systems is the input application flow. Applications enter 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. The 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 service of the next request. 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 service requests and the output flow of serviced requests.
The result of the interaction of these flows of random events is the number of applications in the QS at the moment, 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 performance, on the rules for organizing work, has a certain efficiency of functioning (capacity), which allows it to successfully cope with the flow of applications.
Special area of applied mathematics theory of massservice (TMO)– deals with the analysis of processes in queuing systems. The subject of study of the theory of queuing is QS.
The purpose of the queuing theory is to develop recommendations for the rational construction of QS, the rational organization of their work and the regulation of the flow of applications to ensure high efficiency of the QS. To achieve this goal, the tasks of the queuing theory are set, which consist in establishing the dependences of the efficiency of the functioning of the QS on its organization.
The tasks of the queuing theory are of an optimization nature and are ultimately aimed at determining such a variant of the system, which will provide a minimum of total costs from waiting for service, loss of time and resources for service, and from idle service unit. Knowledge of these characteristics provides the manager with information to develop a directed impact on these characteristics to manage the effectiveness of queuing processes.
The following three main groups of (usually average) indicators are usually chosen as characteristics of the effectiveness of the functioning of the QS:
Indicators of the effectiveness of the use of QS:
The absolute throughput of the QS is the average number of requests that the QS can serve per unit of time.
Relative throughput 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.
The average duration of the period of employment of the SMO.
QS utilization rate - the average share of time during which the QS is busy servicing applications, etc.
Application service quality indicators:
Average waiting time for an application in the queue.
Average residence time of an application in the CMO.
Probability of request being denied service without waiting.
The probability that an incoming request will be immediately accepted for service.
The law of distribution of the time the application stays in the queue.
The law of distribution of the time spent by an application in the QS.
The average number of applications in the queue.
The average number of applications in the QS, etc.
The performance indicators of the pair "QS - consumer", where "consumer" means the entire set of applications or some of them
For CMO with failures:
For CMO with unlimited waiting both absolute and relative throughput lose their meaning, since each incoming request will be served sooner or later. For such a QS, important indicators are:
For CMO mixed type both groups of indicators are used: both relative and absolute bandwidth, and expectation characteristics.
Depending on the purpose of the queuing operation, any of the above indicators (or a set of indicators) can be selected as a performance criterion.
analytical model QS is a set of equations or formulas that make it possible to determine the probabilities of system states in the course of its operation and calculate performance indicators based on 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 represent real systems are, as a rule, complex and difficult to see.
Analytical modeling of the QS is greatly facilitated if the processes occurring in the QS are Markovian (the flows of requests are simple, the service times are exponentially distributed). 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, determine the selected performance indicators.
Let's consider 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 without stopping. The flow of trucks is the simplest, the check time is random with an exponential distribution.
Such a situation can be simulated by a three-channel QS with failures (without a queue). The system is open, with homogeneous applications, 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 graph of system states is shown in fig. 2.11.
Rice. 2.11.
On the graph: - the intensity of the flow of trucks; - the intensity of document checks by one traffic inspector.
The simulation is carried out in order to determine the part of the cars that will not be tested.
Solution
The desired part of the probability is the probability of employment of all three inspectors. Since the state graph represents a typical scheme of "death and reproduction", we will find using dependencies (2.2).
The throughput of this post of traffic inspectors can be characterized relative throughput:
Example 2.6. To receive and process reports from the reconnaissance group, a group of three officers was assigned to the reconnaissance department of the association. The expected rate of reporting is 15 reports per hour. The average processing time of 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 if the assigned group of three officers is sufficient to complete the assigned task.
Solution
A group of officers works as a CMO with failures, consisting of three channels.
The flow of reports with intensity 
can be considered the simplest, since it is the total of several reconnaissance groups. Maintenance intensity 
. The distribution law is unknown, but this is not essential, since it is shown that for systems with failures it can be arbitrary.
The description of the 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 the limiting state probabilities for it:
The relation is called the reduced intensity of the flow of applications. Its physical meaning is as follows: the value is the average number of requests coming to the QS for the average service time of one request.
In the example 
.
In the considered QS, failure occurs when all three channels are busy, that is, . Then:
Because failure probability in the processing of reports is more than 34% (), then it is necessary to increase the personnel of the group. Let us double the composition of the group, that is, the QS will now have six channels, and calculate:
Thus, only a group of six officers will be able to process incoming reports with a probability of 95%.
2.5.2. Multichannel QS with waiting
Example 2.7. There are 15 crossing facilities of the same type in the river forcing section. The flow of vehicles arriving at the crossing averages 1 unit/min, the average time of crossing one unit of equipment is 10 minutes (taking into account the return of the crossing facility).
Evaluate the main characteristics of the crossing, including the likelihood of an immediate crossing immediately upon the arrival of a piece of equipment.
Solution
Absolute Bandwidth, i.e. everything that comes to the crossing is almost immediately crossed.
Average number of operating crossing facilities:
Crossing utilization and downtime ratios:
A program was also developed to solve the example. The time intervals for the arrival of equipment at the crossing, the time of the crossing are taken to be distributed according to an exponential law.
The ferry utilization rates after 50 runs are practically the same: 
.
INTRODUCTION
CHAPTER I. FORMULATION OF PROBLEMS OF QUUE SERVICE
1.1 General concept of queuing theory
1.2 Modeling of queuing systems
1.3 QS state graphs
1.4 Stochastic processes
Chapter II. EQUATIONS DESCRIBING QUEUING SYSTEMS
2.1 Kolmogorov equations
2.2 The 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 Multichannel QS with denial of service
3.3 Model of a multi-phase tourist service system
3.4 Single-channel QS with limited queue length
3.5 Single-channel QS with unlimited queue
3.6 Multichannel QS with limited queue length
3.7 Multichannel QS with unlimited queue
3.8 Supermarket queuing system analysis
CONCLUSION
Introduction
At present, a large amount of literature has appeared that is directly devoted 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 the theory of queuing 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 control systems in which homogeneous events are repeated many times, for example, in consumer services 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 and others.
The subject of queuing theory is to establish relationships between the nature of the flow of requests, the number of service channels, the performance of a single channel and efficient service in order to find the best ways to control these processes. The tasks of the queuing theory are of an optimization nature and ultimately include the economic aspect of determining such a variant of the system, which will provide a minimum of total costs from waiting for service, loss of time and resources for service, and from downtime of service channels.
In commercial activities, the application of the theory of queuing has not yet found the desired distribution.
This is mainly due to the difficulty of setting goals, the need for a deep understanding of the content of commercial activities, as well as reliable and accurate tools that allow calculating various options for the consequences of managerial decisions in commercial activities.
Chapter I . Setting queuing tasks
1.1 General concept of queuing theory
The nature of queuing, 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, a mass of commodities from the sphere of production to the sphere of consumption. Such operations are loading of goods, transportation, unloading, storage, processing, packaging, sale. 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, customers, 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. Queues cause a lot of trouble, 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 queuing organization arise in almost all spheres of human activity, for example, servicing buyers in stores by sellers, serving visitors at public 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 tasks can be successfully solved using methods and models of the queuing theory (QMT) specially created for these purposes. This theory explains that it is necessary to serve someone or something, which is defined by the concept of “request (requirement) for service”, and service operations are performed by someone or something called service channels (nodes). The role of applications 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, merchandisers, loaders, commercial equipment, etc. It is important to note that in one variant, 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 for receiving goods.
Due to the massive nature of the receipt of services, applications form flows that are called incoming before servicing operations are performed, and after a possible waiting for the service to begin, i.e. downtime in the queue, form service flows in channels, and then an outgoing flow of requests is formed. In general, the set of elements of the incoming flow of applications, queue, service channels and the outgoing flow of applications forms the simplest single-channel queuing system - QS.
A system is a set of interconnected and. purposefully interacting parts (elements). Examples of such simple QS in commercial activities are places of receipt and processing of goods, settlement centers with customers in shops, cafes, canteens, jobs of an economist, accountant, merchant, cook at distribution, 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 service request request, the state of the service system itself and the service channel.
Indeed, the duration of the buyer's stay in the 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 attendants, which can significantly affect the time spent by the buyer in the supermarket and service intensity. For example, cashier-controllers mastering the "blind" method of working on a cash register made it possible to increase the throughput of settlement nodes by 1.3 times and save time spent on settlements with customers at each cash register by more than 1.5 hours per day. The introduction of a single settlement node in the supermarket gives tangible benefits to the buyer. So, if with the traditional form of settlements, the service time for one customer averaged 1.5 minutes, then with the introduction of a single settlement node - 67 seconds. Of these, 44 seconds are spent on making a purchase in the section and 23 seconds are spent directly on payments for purchases. If the buyer makes several purchases in different sections, then the loss of time is reduced by 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, the requests received need to be serviced by some device. In some cases, the service is performed by one person (customer service by one seller, in some cases by a group of people (patient service by a medical commission in a polyclinic), and in some cases by technical devices (sale of soda water, sandwiches by vending machines). A set of tools that service applications , is called a service channel.
If the service channels are capable of satisfying the same requests, then the 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 successive arrival of customers into the queuing system is called the incoming stream of customers, and the sequence of customers leaving the queuing system is called the outbound stream.
