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Hospital queuing model
Liaoning University of Petrochemical Technology Modeling Group
It is a very common phenomenon to queue up for medical treatment in the hospital. It appears in front of us in one form or another every day. For example, when patients go to the hospital for medical treatment, to the pharmacy for dispensing drugs, and to the infusion room for infusion, they often need to wait in line to receive certain services.
The nurses' desk, charging window, infusion nurses' desk and their service personnel here are all service organizations or service equipment. Patients, like the patients in the shop, are collectively called patients. The above queues are tangible, and some queues are invisible. Due to the randomness of patients' arrival, queuing phenomenon is inevitable.
If the hospital adds more service personnel and equipment, it will increase investment or waste idle time; If the number of service facilities is reduced and the waiting time in line is too long, it will bring adverse effects to patients and society. Therefore, hospital administrators should consider how to strike a balance between the two in order to improve service quality and reduce service cost.
The so-called queuing system simulation modeling is to dynamically simulate the structure and behavior of an objective and complex queuing system by computer, so as to obtain the quantitative index results reflecting the essential characteristics of the system, and then predict, analyze or evaluate the behavior effect of the system, and provide decision-making basis for decision makers.
Hospital queuing theory is a science developed to solve the above problems. It is one of the important branches of operational research.
In queuing theory, patients and service organizations providing various services form a queuing system, which is called random service system. These systems can be concrete or abstract.
Queuing system model has been widely used in various management systems, such as operation management, infusion management, medical service, medical technology business, triage service and so on.
Queuing system and its introduction;
The basic structure of queuing system consists of four parts: arrival process (input), service time, service window and queuing rules.
1, arrival process (input) refers to different types of patients coming to the hospital according to various rules.
2. Service time refers to the time rule of patients receiving services.
3. The service window indicates how many service windows can be opened to receive patients.
4. Queuing rules ensure that arriving patients receive services in a certain order.
Common arrival processes include fixed-length input, Poisson input and Hellem input, among which Poisson input is the most widely used in queuing systems.
Poisson input is an input that satisfies the following four conditions:
① stationarity: the probability of the number of patients arriving in a certain time interval is only related to the length of this time and the number of patients;
② No aftereffect: the number of patients arriving in disjoint time intervals is independent of each other;
③ Commonality: it can be treated or operated at the same time 1 patient, and there is no case of more than 2 cases at the same time;
④ Limited: Only a limited number of patients can be reached in a limited time interval, and it is impossible to reach an unlimited number of patients.
The patient population can be infinite or limited; The arrival mode of patients can be single or batch; The interval between successive arrivals can be determined or random; The arrival of patients can be independent of each other or related; The arrival process can be smooth or non-smooth;
The time law of patients receiving services is often described by probability distribution. Common service time distributions include fixed-length distribution, negative exponential distribution and Erlang distribution.
Generally speaking, the service time of a simple queuing system often obeys a negative exponential distribution, that is, the service time of each patient is independent and identically distributed, and its distribution function is
B ( t ) = 1- e - m t (t ≥0)。
Where m > 0 is a constant, representing the average service rate per unit time, and 1/m is the average service time.
The main attribute of the service window is the number of service desks. Its types are: single service desk and multiple service desks.
Multi-service desks are divided into three types: parallel, series and hybrid. The most basic type is multi-service desk parallelism.
It is divided into three categories: loss system, waiting system and hybrid system.
Loss system: When the patient arrives, if all the service desks are unavailable, the patient will not wait and then disappear from the system.
Waiting system: when patients arrive, if there are no service desks, they will wait in line. There are different rules for the order of waiting for service:
① First-come, first-served services, such as seeing a doctor and queuing for medicine;
(2) first come, first served, such as hospital emergency patients;
③ Random service: When the service desk is idle, the waiting patients are randomly selected for service;
Priority service, such as care number.
Mixed system: there are both waiting and losses, for example, patients decide to stay after considering the queue length and waiting time. The number of queues can be single column or multiple columns; Capacity may be limited or unlimited.
The queuing system model can be mainly described by input process (patient arrival time interval distribution), service time distribution and the number of service desks.
According to these characteristics, symbols can be used to classify and represent different models. For example, the above features are listed by symbols in order by using certain symbol rules and separated by vertical lines, that is,
Input Process | Service Distribution | Service Desk Quantity
For example, M|M|S represents a Poisson input, the service time is negatively exponentially distributed, and the number of service stations is s; M|G| 1 represents a queuing system with Poisson input, general service distribution and single server.
The evaluation and optimization of queuing system need to be reflected by certain quantitative indicators.
Main quantitative indicators of queuing system:
There are three main quantitative indexes for establishing the queuing system model: waiting time, busy period and queue length.
(1) Waiting time refers to the time from the patient's arrival at the system to the start of receiving services. Obviously, patients want to wait as short as possible.
Wq is used to indicate the average waiting time of patients in the system. If service time is considered, Ws is used to represent the average stay time of patients in the system (including waiting time and service time). This indicator reflects the work intensity and utilization of the service desk. B is used to indicate the average length of busy hours. Corresponding to the busy period is the idle period, which refers to the length of time that the service desk has been idle. I is used to indicate the average length of the idle period.
⑶ Captain refers to the number of patients in the system (including all patients waiting in line and receiving services).
Ls is used to represent the average queue length. If patients receiving services are not considered, the number of patients waiting in line in the system is called queue length. Lq is used to represent the average queue length.
In addition, the service intensity is expressed by R, and its value is the ratio of effective average arrival rate L to average service rate M, that is, R = L/m. 。
M | M | 1 model
M|M| 1 model is the simplest queuing system model with Poisson input, negative exponential distribution of service time and single server.
Assuming that the patient source and capacity of the system are infinite, patients are arranged in a single queue, and the queuing rule is first come, first served.
Set the probability Pn(t) of n patients in the T system at any time. When the system reaches a steady state, Pn(t) tends to be balanced, which has nothing to do with T. At this time, the system is said to be in a statistical equilibrium state, and Pn is called the steady-state probability in a statistical equilibrium state.
Pn=( 1- r )r n，n = 0， 1，2，…
Where r =l/m represents the ratio of effective average arrival rate l to average service rate m (0 < r < 1).
Several main indexes of M | M | 1 model
(1) Average number of patients in the system (average queue length) Ls
⑵ Average number of patients in line (average queue length) Lq
⑶ The average stay time of patients in the system Ws.
⑶ The average waiting time of patients queuing in WQ.
5] the average length of idle period I
[6] Average length of busy period b
For example, a nuclear magnetic resonance room is equipped with a professional doctor who is responsible for the vibration shooting of nuclear magnetic resonance. It is known that an average of 6 patients come every day, with an average time 1 hour. The patients who come here arrive according to Poisson distribution, and the service time obeys negative exponential distribution, with 8 hours per day. Try to find out:
(1) the probability of doctors working idle;
② The probability that two patients arrive at the MRI room at the same time;
③ The probability of at least 1 person coming to the magnetic resonance room;
④ The average number of inpatients in MRI room;
⑤ Average stay time of patients in MRI room;
⑥ The average number of patients waiting in MRI room;
⑦ Average waiting time of patients to be photographed;
⑧ The average length of magnetic resonance ventricular busy period.
Average arrival rate l = 6/8 = 0.75 person/hour, average service rate m = 1 person/hour, and service intensity r = 0.75/ 1 = 0.75.
① The probability that patients were not photographed in MRI room was P0 =1-r =1-0.75 = 0.25.
That is, the staff has 25% free time.
② The probability that there are two waiting patients in the MRI room is
P2 = ( 1 - r ) r 2 = 0. 14。
③ The probability that there are at least 1 patient waiting in the MRI room is
P = P(n≥ 1)= 1-P0 = 1-( 1-r)= 0.75。
That is to say, 75% of the time, at least 1 patient is waiting in the magnetic resonance room.
④ The average number of inpatients in MRI room is
M | M | C model
M|M|C(C≥2) is a multi-service waiting and queuing system, and its various characteristics and assumptions are basically the same as M|M| 1 model. It is also assumed that C service stations are arranged in parallel, and each service station works independently, with the same average service rate, that is, m1= m1= … = m c = m.
In the state of statistical balance, the service intensity
The main indicators of M | M | C model are:
(1) Average queue length Lq
(2) Ordinary captain
Ls = Lq + Cr。
⑶ The average stay time of patients in the system Ws.
⑶ The average waiting time of patients queuing in WQ.