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Hospital Queuing Model
Liaoning University of Petrochemical Technology Modeling Group
Hospital queuing is a very familiar phenomenon that is often encountered. It appears to us every day in one form or another. For example, when a patient goes to a hospital, when a patient goes to a pharmacy to dispense medicine, or when a patient goes to an infusion room to receive an infusion, he or she often has to wait in line to receive some kind of service.
Here, the nurse's desk, the charge window, the infusion nurse desk and its service personnel are service organizations or service equipment. The patients are the same as those in the store, collectively referred to as patients. The above queues are tangible, there are some queues are intangible. Because of the randomness of the arrival of patients, so the phenomenon of queuing is unavoidable.
If the hospital to add service personnel and equipment, we must increase investment or idle waste; if the service equipment to reduce the waiting time is too long, the patients and society will have a negative impact. Therefore, hospital managers should consider how to strike a balance between the two, in order to improve the quality of service, reduce service costs.
The so-called queuing system simulation modeling, is the use of computers on an objective and complex queuing system structure and behavior of dynamic simulation, in order to obtain the results of the quantitative indicators reflecting the nature of its system, and then predict, analyze or evaluate the behavioral effects of the system, to provide decision makers with a basis for decision-making.
Hospital queuing theory, is to solve the above problems and the development of a science. It is one of the important branches of operations research.
In queuing theory, patients and service organizations that provide various forms of services form a queuing system, called a stochastic service system. These systems can be concrete or abstract.
The queuing system model has been widely used in various management systems. For example, surgery management, infusion management, medical services, medical technology business, triage services, and so on.
Queuing system and introduction:
The basic structure of the queuing system consists of four parts: come to the process (input), service time, service window and queuing rules.
1, to the process (input) refers to different types of patients come to the hospital in accordance with various laws.
2. Service time is the time pattern of patients receiving services.
3. The service window indicates how many service windows are open to accommodate patients.
4, queuing rules to determine the arrival of patients in a certain order to receive services.
Commonly come to the process of fixed-length input, Poisson (Poisson) input, Erlang (A. K. Erlang) input, etc., of which Poisson input in the queuing system is the most widely used.
The so-called Poisson input is the input that satisfies the following four conditions:
① Stability: the probability of the number of patients arriving in a certain time interval is only related to the length of the time period and the number of patients;
② No posteriority: the number of patients arriving at disjointed time intervals is independent of each other;
③ Ordinaryness: the number of patients arriving at most 1 patient at the same point of time, and no patient arriving at most 1 patient at any time point. (iii) Ordinaryness: at most one patient arrives at the same point in time for consultation or surgery, and there is no case of more than two patients arriving at the same time;
(iv) Finiteness: only a finite number of patients can arrive at a finite time interval, and it is not possible to have an infinite number of patients arriving at a finite time interval.
The total number of patients can be infinite or finite; patients can arrive individually or in batches; the interval between successive arrivals can be deterministic or stochastic; patient arrivals can be independent of each other or correlated; the process of arrivals can be smooth or non-smooth;
The temporal pattern of patients' acceptance of services is often described by a probability distribution. are often also described by probability distributions . Common service time distributions are the fixed-length distribution, the negative exponential distribution, and the Erlang distribution.
In general, the service time of a simple queuing system tends to follow a negative exponential distribution, i.e., the service time of each patient is independently and identically distributed, with the distribution function
B ( t ) = 1- e - m t (t ≥ 0).
Where m > 0 is a constant representing the average service rate per unit of time. And 1/m is the average service time.
The main attribute of a service window is the number of service desks. The types are: single service desk, multiple service desks.
Multi-service desks are further divided into parallel, tandem and hybrid types. The most basic type is the parallel connection of multiple service desks.
Divided into three categories: loss system, waiting system, and hybrid system.
Loss system: when a patient arrives, if all the service desks are unavailable, the patient does not want to wait and disappears from the system.
Waiting system: the patient arrives, if all the service desk is not available, they wait in line. Waiting for the order of service and a variety of different rules:
① first-come-first-served, such as medical treatment, queuing for medication, etc.;
② first-come-first-served, such as hospitals to deal with emergencies;
③ randomized service, the service desk is idle, randomly selected patients waiting for service;
④ priority service, such as the care of number.
Mixed system: both waiting and loss of circumstances, such as patients waiting to consider the captain of the queue, the length of time waiting and other factors to decide to stay or go. The number of queues may be single or multiple; the capacity may be finite or infinite
The queuing system model can be characterized by the input process (distribution of patient arrival intervals), the distribution of service times, and the number of service stations.
Based on these characteristics, symbols can be used to categorize and represent different models. For example, the above features are listed in the order of symbols using a certain notation rule and separated by vertical lines, i.e.
Input Process | Service Distribution | Number of Service Desks
For example, M|M|S denotes a queuing system model with Poisson input, negative exponential distribution of service time, and S service desks; M|G|1 denotes a queuing system with Poisson input, generalized service distribution, and single service desk; and M|G|1 denotes a queuing system with Poisson input, generalized service distribution, and a single service desk. queuing system with a single service desk.
The evaluation and optimization of queuing system need to be reflected by certain quantitative indicators.
The main quantitative indicators of queuing system:
There are three main quantitative indicators for modeling queuing system: waiting time, busy period and queue leader.
(1) Waiting time is the period of time from the time a patient arrives at the system until the time he or she begins to receive service. Obviously, patients want to wait as long as possible.
The average waiting time of a patient in the system is denoted by Wq. If service time is taken into account, Ws is the average length of time a patient spends in the system (including both waiting time and service time). This indicator reflects the intensity and utilization of the service desk. The average length of the busy period is denoted by B. Corresponding to the busy period is the idle period, which is the length of time that the service desk remains idle. Use I to denote the average length of the idle period.
(3) Captain refers to the number of patients in the system (including all patients waiting in line and being served).
Denote the average captain by Ls. If patients receiving services are not considered, the number of patients waiting in line in the system is referred to as the queue length. The average queue length is denoted by Lq.
In addition, denote the service intensity by r, whose value is the ratio of the effective average arrival rate l to the average service rate m, i.e., r = l/m .
M | M | 1 Model
The M | M | 1 model is the simplest model of a waiting queueing system with Poisson inputs, negatively exponentially distributed service times, and a single service desk.
Assume that the system has an infinite patient source and capacity, and that patients are arranged in a single queue with a first-come, first-served queuing rule.
Let there be a probability Pn(t) that there are n patients in the system at any time t. The probability Pn(t) is the same as the probability Pn(t). When the system reaches a steady state, Pn(t) tends to equilibrium Pn and is independent of t. At this point, the system is said to be in a statistical state. At this time, the system is said to be in statistical equilibrium, and Pn is said to be the steady-state probability in statistical equilibrium.
Pn=(1- r )r n, n = 0, 1, 2, ... .
Where r = l/m denotes the ratio of the effective average arrival rate l to the average service rate m (0 < r < 1).
M | M | 1 A few key metrics for the model
(1) Average number of patients in the system (average captain) Ls
(2) Average number of patients waiting in the queue (average queue length) Lq
(3) Average length of time a patient stays in the system Ws
(4) Average length of time patients wait in the queue Wq
(5) Idle period The average length of the idle period I
(6) The average length of the busy period B
Example An MRI room is staffed by a physician who specializes in nuclear magnetic **** vibration photography. It is known that the average number of patients per day is 6, the average time per person is 1 hour, the patients arrive according to the Poisson distribution, the service time obeys the negative exponential distribution, and the day is 8 hours. Find:
① the probability that the physician is free;
② the probability that two patients arrive at the same time in the MRI room;
③ the probability that at least one patient arrives in the MRI room;
④ the average number of patients staying in the MRI room;
⑤ the average length of time that a patient remains in the MRI room;
⑥ the average number of patients waiting in the MRI room;
the average number of patients waiting for a patient;
the average number of patients waiting for the MRI room;
the average number of patients waiting for a patient; and average number of patients waiting in the MRI room;
⑦ average waiting time for patients to be photographed;
⑧ average length of the busy period in the MRI room.
Solution The average arrival rate l = 6/8 = 0.75 person/hour, the average service rate m = 1 person/hour, the intensity of service r = 0.75/1 = 0.75.
① The probability that there is no patient in the MRI room for filming is P0 = 1 - r = 1 - 0.75 = 0.25.
That is, the staff is available 25% of the time.
② The probability that there are 2 waiting patients in the MRI room is
P2 = (1 - r ) r 2 = 0.14.
③ The probability that there is at least 1 waiting patient in the MRI room is
P = P (n ≥ 1) = 1 - P0 = 1 - (1 - r ) = 0.75 .
That is, 75% of the time, there is at least 1 waiting patient in the MRI room.
④ The average number of patients staying in the MRI room is
M | M | C model
M | M | C (C ≥ 2) is a multi-desk waiting queuing system, and its various characteristics are specified and assumed to be basically the same as that of the M | M | 1 model. It is also assumed that C service desks are arranged in parallel and each service desk works independently with the same average service rate, i.e., m 1 = m 1 = ... = m C = m . Therefore, the average service rate of the system is Cm .
In statistical equilibrium, the service intensity
M | M | C model main indicators are:
(1) Average queue length Lq
(2) Average captain Ls
Ls = Lq + Cr .
(3) Mean patient stay in the system Ws
(4) Mean patient wait time in the queue Wq