4.2.1
Literature Review
Figure 2.2 illustrates a weekly average profile of elective patients admissions to the CCU. Unlike emergency arrivals, elective demand is dependent on day of the week. The majority of elective admissions occur on Thursday with very few on the weekend.
Whilst much literature is devoted to the analysis of a service system with constant arrival and ser- vice times (Green and Kolesar, 1991 [66]; Pollaczek, 1934 [135]), most actual systems are subject to time-varying demand, where arrival rates and the number of servers vary throughout the period of operation. Computer systems, road traffics, telecommunications networks, banks, airports, toll booths and hospitals systems are just a few examples of facilities with time-varying demand pro-
Chapter 4 FURTHER APPLICATIONS OFMATHEMATICAL MODELLING 82 cesses. The steady-state theory developed by Erlang was inadequate for time-dependent systems, and began numerical investigations into the behaviour of the system during a finite interval. The study of time-dependent queues remains a vibrant area of research. The process involved are far more complex and as a consequence more sophisticated mathematical procedures are necessary (Channouf at al., 2007 [24]; Holcomb and Sharpe, 2007 [87]; Feldman et al., 2008 [51]; Bekker and de Bruin, 2010 [13]; Caiado, 2010 [23]; Izady and Worthington, 2012 [92]). Analytical models for such situations are often intractable, but in addition to numerical approaches, approximation methods have been developed, that provide reliable results in suitable scenarios (Green et al., 1991 [66]).
Several approximation methods have been proposed in the literature which use series of tractable stationary models to estimate the time-dependent nature of a system. These methods, however only give reliable results under certain conditions. They do not consider non-stationary and transient effects, so will only be accurate if the rate of change of the arrival rate relative to the throughput of the system is sufficiently low to allow the system to quickly achieve the steady state associated with any arrival rate (Utley and Worthington, 2011 [158]).
Two methods, which make use of compartmentalized steady-state models to find the minimum number of servers required to meet a desired service target in each planning period are the stationary independent period-by-period approach (SIPP) and the pointwise stationary approximation (PSA). Whilst variants of these methods have been developed in the literature over the last four decades (Kolesar et al., 1975 [105]; Green and Kolesar, 1991 [66], 1997 [68]; Green et al., 1991 [67], 2001 [69], 2006 [71]; Ingolfson et al., 2007 [90]), an alternative method known as the modified-offered- load (MOL) has also been investigated (Massey and Whitt, 1994 [119], 1997 [120]; Ingolfsson et al., 2007 [90]).
Whilst approximation methods provide instant solutions, the numerical approaches are able to offer solutions with a higher degree of accuracy at the expense of computational time.
In the literature, there exist four numerical approximations to the solution of differential equations; thus the M (t)/M (t)/c(t) time-dependent equations can be solved using one of the following meth- ods:
• Euler’s method; • Runge-Kutta method; • the randomization method;
Chapter 4 FURTHER APPLICATIONS OFMATHEMATICAL MODELLING 83 Further details regarding the theoretical underpinnings of the numerical methods used to solve the time-dependent equations is provided in Gross and Harris, 1998 [77].
The randomization method (described in Grassmann, 1977 [63]) is a method to compute transient solutions of finite state continuous-time Markov chains. The method involves the constructions of an analogous discrete time Markov chain, where transitions occur according to an exponential distribution with the same parameter in every state. The method provided similar results as the Runge-Kutta (Ingolfsson et al., 2007 [90] ), but was more computationally efficient.
The DTM method produces accurate results at a much faster time than several approximation meth- ods (Wall and Worthington, 2007 [165]). The approach uses discrete-time models to approximate the behaviour of continuous time queues by dividing the time of operation of the system into a set of non-overlapping intervals. The technique creates a transition matrix to take account of the various states that occur at each time step, and evaluate the probabilities associated with each state. Euler’s and Runge-Kutta methods are general approaches for solving ordinary differential equa- tions. The Runge-Kutta approach provides solutions that are referred to as ’exact’ since the only approximations required are the approximation of the infinite set of equations with a finite set, and those inherent in any numerical solution of ordinary differential equations. Euler’s method, how- ever has the advantage that it may be implemented to provide solutions at a quicker rate and does not require an ordinary differential equation solver (Izady, 2010 [91]).
Euler’s method considers the slope of the tangent line to approximate the solution at each inter- val. It approximates the solution by evaluating the equations at a starting value, and then at steps separated by small time intervals δt (between which the solution is not expected to have changed greatly). Smaller step sizes generate solutions with higher accuracies, but this comes at a greater computational cost (Izady, 2010 [91]). This method is investigated in the form of case study in Section 4.2.2.
4.2.2
Time-Dependent Bed Utilisation
It was noted in Section 2.3.1 that the admission of patients undergoing elective surgery depends on day of the week. The importance of incorporating the time-dependent nature of these arrivals has been highlighted (Costa et al., 2003 [35]). It was noted that the difference in the arrival rates during weekdays is not as visible as between weekdays and weekend. Therefore this section will adapt the mathematical model to take this factor into account and allow for different elective arrival rates for weekdays and weekend.
Chapter 4 FURTHER APPLICATIONS OFMATHEMATICAL MODELLING 84 The differential-difference equations given in the set of Equations 3.1 are solved using a numerical iterative method (Euler’s method), described in Section 4.2.1. A step-length of δt = 0.01 (hours) was used, and the process is initially run to produce bed occupancy levels for 1 week (168 hours). The iterative process is implemented in Visual Basic, linked to an Excel spreadsheet, which enables key variables to be altered quickly and easily. The Visual Basic program calculates the required probabilities, and outputs these to a table in Excel. The mean and the standard deviation of the number of beds occupied are also calculated and included in the results.
Consider the results evaluated when using the parameters obtained from the data and which are given in Table 4.1.
Table 4.1: Parameter values
Parameter Parameter name Parameter Value c Number of service channels (beds) 29
λ1 Emergency arrival rate (per day) 2.6081
λ2 Elective arrival rate at weekdays (per day) 1.5808
λW
2 Elective arrival rate at weekends (per day)) 0.3542
µ1 Emergency service rate (days) 0.1411
µ2 Elective service rate (days) 0.3262
δt Time increment (hours) 0.01
The appropriate values of λ2, are used for the first 120 hours of the week (i.e. Monday to Friday),
and then the reduced value for λW2 are used over the period 120-168 hours (Saturday and Sunday). Initially (at time t = 0), P0, the probability of having an empty system is set equal to 1 with all
other initial probabilities (Pn, where n = 1, . . . , 29) being set to 0. In this way, it is assured that the
initial assignment of probabilities sums to one. This process is repeated and a new set of probabili- ties are generated with every increment of δt, using the probabilities at time t to calculate the new probabilities at time t + δt. At every time interval probabilities are output to a spreadsheet, so that their behaviour over time can be monitored.
As mentioned previously, the program is initially run for 1 week (168 hours), but to make sure all probabilities get to steady-state solution, it should be run for a longer period of time. The program is therefore run for 4 weeks (672 hours) and results from the last week are shown in Figure 4.1. For illustration purposes, only a few selected probabilities are shown. As expected, over the weekend (120-168 hours), the probabilities of lower bed occupancy increase (e.g. P8, P16 and P20), and the
Chapter 4 FURTHER APPLICATIONS OFMATHEMATICAL MODELLING 85
Figure 4.1: Probabilities of beds occupancies over the week
It is also of an interest how the mean and standard deviation of the bed occupancy change over the week, as shown in Figure 4.2.
Figure 4.2: Variation in bed occupancy over the week
The blue line in Figure 4.2 indicates the mean of bed occupancy, while the red line indicates the standard deviation. As expected, the mean bed occupancy decreases on the weekend (from 22.36 to 19.40), but surprisingly, the variation increases (from 3.93 to 4.17).
The effect of increasing elective admissions on the weekend is also investigated. Very interestingly, if it is increased by 1 patient per day on the weekend, the mean and standard deviation of the bed occupancy remain almost the same throughout the whole week; it decreases from 21.52 to 22.22 and the standard deviation increases from 4.002 to 3.954.
Chapter 4 FURTHER APPLICATIONS OFMATHEMATICAL MODELLING 86 Going one step further, and increasing elective admissions on the weekend by 1.5 patients per day would increase the mean bed occupancy by over 1 bed (from 21.52 to 22.51) and decrease the stan- dard deviation from 4.002 to 3.91.
The transient times (times required to get from one steady-state to another steady-state level) are also obtained. If the incremented difference is less than 0.01% it is assumed that the system is at steady-state. The overall transient time was the lowest (approximately 20 hours) for bed occupancy levels between 19 and 22, which confirms what was said earlier, that the probability of having 20 beds occupied remains almost the same throughout the whole week.
The quantitative information presented is of obvious importance to the CCU Director in informing their decision making regarding bed management over the seven day cycle. The next section will investigate forecasting of future bed occupancies with probabilistic models.