Keywords Queuing theory; stochastic model; hospital management
Hospitals are a complex system, patients from registration, consultation, pricing, pick up medication for each service, when the existing demand for a particular service exceeds the existing capacity to provide that service, queuing occurs, due to the stochastic nature of the time of the patient's arrival and the time required to diagnose and treat the patient, little controllability, queuing is almost Unavoidable, when the clinic is not enough, patients often wait in line for too long, patient satisfaction decreases, medical staff work is too busy, easy to make mistakes caused by doctor-patient disputes, patients and society will bring adverse effects. Therefore, how to rationalize the scientific arrangement of medical personnel and their medical equipment, so that the hospital will not blindly increase the number of doctors and equipment caused by unnecessary idle, the formation of a waste of resources, but also to minimize the waiting time of patients in line, how to strike a balance between the two, in order to improve the quality of service, reduce the cost of services, which is the modern hospital administrators have to face the problem.
The queuing theory model (quening theory model), is quantitatively through mathematical methods, the structure and behavior of an objective complex queuing system for dynamic simulation research, scientific and accurate description of the probability of the queuing system law, queuing theory is also an important sub-discipline of operations research [1, 2]. In hospital management, if on the basis of queuing theory, the structure and behavior of the queuing system of hospital outpatient clinics and consultation rooms are scientifically simulated and systematically studied. Thus, the optimal design of clinic and doctor arrangements to obtain the results of quantitative indicators reflecting the essential characteristics of its system, for prediction, analysis or evaluation, to maximize the satisfaction of the needs of patients and their families will be effective in avoiding the waste of resources.
1 Stochastic model
1.1 System description
Taking the hospital outpatient clinic as the object of study, it has the following characteristics:
① Input process:The arrival of patients is independent of each other, and the interval between successive arrivals is randomized; arrivals at a certain time obey the Poisson distribution.
② queuing rules: from first-come-first-served, and for the waiting system, that is, when the patient arrives at all clinics and doctors are not available, they have to wait in line.
3 service time: patient consultation time is independent of each other, obeying the negative exponential distribution.
④ Service window: multiple service desks, C service desks arranged in parallel, the service desk independent work.
1.2 Model Assumptions and Establishment
Assuming that the average arrival rate of patients is λ, the average service rate of a single service desk (which indicates the number of patients who have been served per unit of time) is μ, and the average service rate of the entire service organization is cμ; the service intensity of the system ρ = λ/cμ<1 before it will not be lined up into an infinite queue (the average utilization rate of the service desks), and that pn(c) is the average rate of C probability that there are n patients in the system at any moment of the service desk; when the birth and death process with an arrival rate of λ and a service rate of cμ reaches a steady state, it can be obtained:
p0(c)=[∑c-1k=01k!(λμ)k+1c! 1(1-ρ) (λμ)c]-1(1)
pn(c)=1n!(λμ)np0(c), n=1,2,. ...,c
1c!cn-c (λμ)np0(c), n=c+1,...(2)
When the system reaches equilibrium, the mean value of each patient's waiting time W in the system is:
E(W)=pn(c)cμ(1-ρ)2=nμn!(n μ-λ)2 (λμ)np0(c)(3)
The number of people staying in the queue Ls=Lq+cρ=1c! (cρ)cρc!(1-ρ)2p0+λμ(4)
1.3 Optimization of queuing system
In a queuing system, the patient hopes that the more service desks, the more efficient the service, and the shorter the staying time the better, so as to minimize his loss. To minimize the hospital to increase the number of doctors and equipment, and the hospital can not be infinite investment. Therefore, it is necessary to optimize the design, the purpose of which is to minimize the sum of the patient's loss and the hospital's service cost. Assuming that the number of service desks is c, cs is the cost of each service desk per unit of time service desk fee, cw is the cost of each patient to stay in the system per unit of time, the total cost of Z (c) (the expected value of the total cost per unit of time, which is a function of the number of service desks for c), then the objective function minz (c) = Csc + CwLs (c) , where Ls is the number of people who stayed (Equation (4)). c can only take an integer, set c* is to make the objective function c to take the minimum value of the point, c * meet
z (c*-1) ≤ z (c*) = csc* + cwLs (c*) ≤ z (c* +1), Ls = Ls (c)
Simplified to Ls (c*) - Ls (c* +1) ≤ cscw ≤ Ls (c* -1) - Ls (c*)(5)
Simplify to Ls (c*) - Ls (c*)(5)
The objective function of c is to take the minimum value of c, where Ls is the number of stay (Equation (4)). p>
Through computer simulation to calculate Ls(1), Ls(2), Ls(3), ... the difference between two adjacent items in turn, to see the constants fall between which two, so as to determine so that the sum of the cost of the patient's loss and the cost of hospital services to optimize the optimal number of service stations c optimal solution C *.
1.4 Optimization of the Service Solution Problem
When the average patient arrival rate increases causing an increase in service intensity resulting in the average captain L being too large, or even the captain tends to be infinite due to the service intensity ρ>1, the only option is to increase the number of service desks while keeping the average service rate constant. The following is a discussion of the case where there are 2 service stations and their average service rates are equal.
There are two forms of queuing service for 2 service desks as shown in the following two figures:
Figure 1 is an M/M/2 model with only one queue, and figure 2 is a 2 M/M/1 model with two queues and no switching of queues after entry.
Figure 1 (omitted)
Figure 2 (omitted)
We can know that the two service desks of the two forms of service average captain L, the ratio of the waiting time W is:
2L1L2 = W1W2 = 1 + ρ2 (ρ2 = λ2μ<1)
On the waiting time that people are most concerned about there is a 1
The same reasoning can be proved: in a queuing system with multiple side-by-side waitstations, queuing into a single queue has a significant superiority over the scheme of queuing into multiple parallel queues. For a stochastic process with multiple attendants, patients should be placed in a single queue if the waiting time is the only consideration.
2 Example analysis
A hospital operating room in order to grasp the randomized service situation, statistics of 100h patient visits and completed surgery data, as shown in the following table: (omitted)
① Calculate the corresponding quantitative indicators;
② If the hospital would like to build an operating room of the same size, ask whether it is reasonable?
With the help of MATLAB software:
1) Firstly, calculate the average arrival rate of patients per h λ=∑nfn/100=210/100=2.1 (h/person), the average time of surgery 1/μ=∑vfv/100=40/100=0.4 (person/h), the number of people completing surgery per hour μ=1/0.4=2.5; with the goodness-of-fit of the χ2=∑6n=0(fn-100pn)100pn to test whether the mean arrival rate λ=2.1 conforms to the Poisson distribution;
Calculating χ2=3.06, taking α=0.05 to get the critical value χ2α=11, because χ2α=11>χ2=3.06, so the acceptance of the arrival rate obeys the Poisson distribution with parameter λ=2.1. . Similarly it can be tested that the operation time obeys the exponential distribution with parameter 2.5. The main quantitative indicators of the queuing system using the above formula are as follows;
Number of patients in the system 5.25 ( people) Number of patients waiting in the queue 4.41 (people) Patient sojourn time 2.5 (h) Queuing waiting time 2.1 (h) Service intensity ρ = λ / μ = 0.84 Coefficient of loss of patient's time 5.25 Probability of idle time in the operating room 0.16 Probability of busy time pn = 0.84
② Calculation Service intensity ρ=λ/cμ=0.42<1 Quantity metrics with the addition of an operating room of the same size
Number of patients in the system 1.02 ( people) Number of patients waiting in queue 0.18 (people) Patient length of stay 0.48 (h) Queuing time 0.08 (h) Probability of two ORs being idle 0.4 Only the probability of Probability that one operating room is free p1=0.34 Probability that a patient doesn't have to wait 0.74 Probability that a patient has to wait 0.26
Based on the above data indicators, it can be concluded that: there is only one operating room in the department where the patient's waiting time is 5.25 times longer than the operating time; 84% of the time in the operating room is busy, and only 16% of the time is free. If another operating room is added the probability of it being utilized is 42%, the probability of it being idle is 58%, the probability of two operating rooms having idle time is 0.4, and the probability of only one of the two operating rooms being idle is 34%. Based on the above data decision makers can decide whether to add an operating room, thus providing managers with a tool for decision support.
3 Conclusion
The queue to the hospital is a commonplace phenomenon, due to the randomness of the patient's arrival and the time of medical services, the number of patient sources is theoretically unlimited, and medical resources are limited, how to make qualitative under the limited resource allocation, using the above queuing model theory and computer simulation, combined with the patient's service records obtained from the relevant data, quantitative quantitative indicators, and then predict, analyze and evaluate, through the optimization of design, the implementation of dynamic management, according to the strength of the hospital, improve facilities and equipment, a reasonable increase in the number of medical and nursing staff, improve the level of diagnosis and treatment technology of doctors, effectively shorten the average consultation time and its fluctuation degree, improve efficiency, shorten the waiting time, unify the diagnostic and treatment procedures, and provide relief for patients. Obviously, the application of queuing theory, on the one hand, can effectively solve the problem of personnel and equipment configuration in the hospital service system, to provide a reliable decision-making basis for hospital management; on the other hand, through the optimization of the system, to find out the balance between the patient and the hospital, not only to reduce the waiting time of the patient queuing, but also not a waste of human and material resources of the hospital, so as to obtain the maximum social and economic benefits.
References
1 Han Botang. Management Operations Research. Beijing: Higher Education Press, 2005, 307-322.
2 Jiang Qiyuan. Mathematical Modeling. Beijing: Higher Education Press, 1993, 456-467.
3 Bian Fuping, Hou Wenhua, Liang Fengzhen. Mathematical modeling methods and algorithms. Beijing: Higher Education Press, 2005, 262~276.
4 Bian Fuping,Hou Wenhua,Liang Fengzhen.