Question

In capacity planning in healthcare, there exists a nonlinear relationship between average delay, utilization, system variability and system size.

(a) Explain those relations clearly.

(b) Prove the following relation by using queuing theory.

If the size of the system increases, the average delay decreases for a given utilization level. In other words, queueing systems have economies of scale.

Answer 1

(a) Capacity planning in health care is greatly complicated by the nature of the service. The individuality of each patient leads to a multiplicity of potential resource requirements often consumed in sequence and with varying degrees of urgency. Patients are also able to refuse treatment, lobby for a different type of treatment, or simply fail to show up for treatment. Finally, the dire consequences sometimes resulting from untimely access lead to much lower tolerances with regards to meeting performance targets. All these complexities make for significant challenges when seeking to plan capacity in the health care setting.

1. Average Delay: It referred to the number of days between the onset of symptoms and the first consultation with the doctor.

2. Utilization: It is the use of managed care techniques such as prior authorization that allow payers, particularly health insurance companies to manage the cost of health care benefits by assessing its appropriateness before it is provided using evidence-based criteria or guidelines.

3. System Variability: It is in the use of effective care can be due to differences in clinical knowledge, differential rates of diffusion and adoption of innovation.

4. System Size: It refers to the allocation provided within the healthcare to hold and run of maximum capacity to fulfil the demand and necessity with at most proficiency

(b) Queueing theory deals with system performance in steady-state. That is, most queueing models assume that the system has been operating with the same arrival, service time and other characteristics for a sufficiently long time that the probability distribution for the queue length and customer delay is independent of time. Clearly, there are many service systems, including health care systems, for which there are time-of-day, day-of-week or seasonality effects.

In queueing theory, utilization, defined as the average number of busy servers divided by the total number of servers times 100, is an important measure. From a managerial perspective, utilization is often seen as a measure of productivity and therefore it is considered desirable for it to be high. For example, in hospital bed planning, utilization is called occupancy level and historically, an average hospital occupancy level of 85 per cent has been used as the minimum level. Since the actual average occupancy rate for nonprofit hospitals has recently been about 66 per cent, there has been a widely held perception in the health care community that there are too many hospital beds. Largely because of this perception, the number of hospital beds has decreased by almost 25 per cent in the last 20 years.

But determining bed capacity based on occupancy levels can result in very long waiting times for beds. In all queueing systems, the higher the average utilization level, the longer the wait times. However, it is important to note that this relationship is nonlinear. There are three critical observations we can make. First, as average utilization (e.g. occupancy rate) increases, average delays increase at an increasing rate. Second, the average delay increases dramatically in response to even small increases in utilization. Finally, the average delay approaches infinity as utilization approaches one.

Variability generally exists in both the time between arrivals and the duration of service times and is usually measured by the ratio of the standard deviation to the mean, called the coefficient of variation (CV). System size is defined as the ratio of the average demand over the average service time, which is a determinant of the number of servers needed.

These basic queueing principles have several important implications for planning or evaluating capacity in a service system. First, the average total capacity, defined as the number of servers times the rate at which each server can serve customers, must be strictly greater than the average demand. In other words, unless average utilization is strictly less than 100%, the system will be “unstable” and the queue will continue to grow. Though this fact may appear counter-intuitive on the surface, it has been well known by operations professionals for decades. So if an emergency room has 10 patients arriving per hour on average and each healthcare provider (physician or physician assistant) can treat 2 patients per hour, a minimum of 6 providers are needed. (Of course, in many contexts, if arrivals see a long queue they may not join it or they may renew after waiting a long time. If so, it may be possible to have stability even if the average demand exceeds the average capacity.) Second, the smaller the system, the longer the delays will be for a given utilization level. In other words, queueing systems have economies of scale so that, for example, larger hospitals can operate at higher utilization levels than smaller ones yet maintain similar levels of congestion and delays. Finally, the greater the variability in the service time (e.g. length-of-stay), the longer the delays at a given utilization level. So a clinic or physician office that specializes in e.g. vision testing or mammography, will experience shorter patient waits than a university-based clinic of the same size and with the same provider utilization that treats a broad variety of illnesses and injuries.