Queueing network model for obstetric patient flow in a hospital

Abstract

A queueing network is used to model the flow of patients in a hospital using the observed admission rate of patients and the histogram for the length of stay for patients in each ward. A complete log of orders for every movement of all patients from room to room covering two years was provided to us by the Medical Information Department of the University of Tsukuba Hospital in Japan. We focused on obstetric patients, who are generally hospitalized at random times throughout the year, and we analyzed the patient flow probabilistically. On admission, each obstetric patient is assigned to a bed in one of the two wards: one for normal delivery and the other for high-risk delivery. Then, the patient may be transferred between the two wards before discharge. We confirm Little’s law of queueing theory for the patient flow in each ward. Next, we propose a new network model of M/G/ ∞ and M/M/ m queues to represent the flow of these patients, which is used to predict the probability distribution for the number of patients staying in each ward at the nightly census time. Although our model is a very rough and simplistic approximation of the real patient flow, the predicted probability distribution shows good agreement with the observed data. The proposed method can be used for capacity planning of hospital wards to predict future patient load in each ward.

Appendix: Relevant queueing models

The following queueing models are relatively simple and robust, and the explicit formulas are available for the probability distribution of the number of customers present in the system.

• (1) M/G/ ∞

  A model denoted by M/G/ ∞ in Kendall’s notation of queueing theory is simply a system with sufficiently many servers to which customers arrive in a Poisson process and spend a random amount of service time, which is generally distributed probabilistically [9, p.84], [11, p.145]. There is no contention for servers among customers. If λ denotes the arrival rate and b denotes the mean service time, the number N of customers present in the system at an arbitrary time has a Poisson distribution with mean ρ: = λ b:

  $$ P \{ N = k \} = \frac{{ \rho^{k} }}{{ k! }} e^{- \rho} \qquad k \ge 0 . $$

  (27)

  Note that this distribution depends on the service time only through its mean value.

  A useful property in modeling is that the output of an M/G/ ∞ system is a Poisson process [18]. A nice property about the Poisson process is that the superposition of independent Poisson processes forms another Poisson process with added rates. These properties contribute to building a simple and robust model.

• (2) M/M/ m

  A model denoted by M/M/ m in Kendall’s notation is a queueing system with m servers and a waiting room of infinite capacity to which customers arrive in a Poisson process at rate λ each with the service time exponentially distributed with mean 1/μ [9, p.66], [11, p.142]. Then, the probability distribution for the number N of the customers present in the system at an arbitrary time in the steady state is given by

  $$ P \{ N = k \} = \left\{\begin{array}{ll} P_{0} (m , \rho ) \frac{{ \rho^{k} }}{{ k! }} & \quad 0 \le k \le m , \\ P_{0} (m , \rho ) \frac{{ m^{m} }}{{ m! }} \left( \frac{ \rho}{m } \right)^{k} & \quad k \ge m + 1 , \end{array}\right. $$

  (28)

  where ρ: = λ/μ and

  $$ \frac{1}{{ P_{0} (m , \rho ) }} = \sum\limits_{k=0}^{m-1} \frac{{ \rho^{k} }}{{ k! }} + \frac{{ \rho^{m} }}{{ (m - 1 )! (m - \rho ) }} . $$

  (29)

  The output of an M/M/ m system is also a Poisson process [3].

  Each customer in the system is either waiting in the waiting room or being served. The probability distribution and the mean for the number L of customers in the waiting room are given by

  $$\begin{array}{@{}rcl@{}} P \{ L = k \} &=& P \{ N = m + k \} \\ &=& \left\{\begin{array}{ll} 1 - \frac{ \rho}{m } C (m , \rho ) & k = 0 , \\ C (m , \rho ) \left( 1 - \frac{ \rho}{m } \right) \left( \frac{ \rho}{m } \right)^{k} & k \ge 1, \end{array}\right. \\ E [ L ] &=& \frac{{ \rho C (m , \rho ) }}{{ m - \rho }} , \end{array} $$

  (30)

  where the Erlang’s C formula [9, p.70]

  $$ C (m , \rho ) := \frac{{ \frac{{ \rho^{m} }}{{ m! }} }}{{ \left( 1 - \frac{ \rho}{m } \right) \sum\limits_{k=0}^{m-1} \frac{{ \rho^{k} }}{{ k ! }} + \frac{{ \rho^{m} }}{{ m ! }} }} $$

  (31)

  gives the probability that an arriving customer waits because all servers are busy.

  The probability distribution and the mean for the waiting time (the time that a customer spends in the waiting room) W in the M/M/ m queue are given by

  $$\begin{array}{@{}rcl@{}} P \{ W = 0 \} &=& 1 - C (m , \rho ) , \\ P \{ W < t \} &=& 1 - C (m , \rho ) e^{- (m - \rho ) \mu t } \qquad t > 0 , \\ E [ W ] &=& \frac{{ C (m , \rho ) }}{{ \mu (m - \rho ) }} . \end{array} $$

  (32)

  We note that the relation E[L] = λ E[W] is an example of Little’s law in Eq. 12.

  The probability distribution and the mean for the number S of customers in service are given by

  $$\begin{array}{@{}rcl@{}} P \{ S = k \} &=& \left\{\begin{array}{ll} P \{ N = k \} & 0 \le k \le m - 1 \\ P \{ N \ge m \} & k = m , \end{array}\right. \\ &=& \left\{\begin{array}{ll} P_{0} (m , \rho ) {{ \rho^{k} } \over { k! }} & 0 \le k \le m - 1 \\ C (m , \rho ) & k = m , \end{array}\right. \\ E [ S ] &=& \rho , \end{array} $$

  (33)

  where P ₀(m,ρ) and C(m,ρ) are given in Eqs. 29 and 31, respectively.