I was putting together a training package today on Queuing Theory for the healthcare industry and the more I got into it, the more I thought it would be a good subject for posting. So today, that's what I'm going to be discussing here on my blog.

I will say that much has been written about queuing theory over the years, but only relatively recently have the healthcare professionals embraced the potential benefits of applying queuing techniques. The theory is actually quite easy to grasp and there are even Excel programs on the net that make the calculations quite easy. Things like staffing numbers, patient arrival rates and service times represent the key data to collect on processes when trying to apply these queuing techniques. The data doesn't even have to be as super accurate as you might expect to run "what if" scenarios. A queuing system is best described as patients arriving for service, waiting for the service if they aren't seen right away, having the services performed and then leaving after they have received the service.

When you're involved in queuing situations, it is necessary to at least estimate the probability distribution or the pattern of the arrival times between successive patient arrivals (i.e. inter-arrival times). When the patients do arrives, they have several choices:

- They can wait no matter how long the wait time estimate is
- They can "balk" or leave immediately because they feel the queue is too large
- They can "renege" or leave after waiting for some time

It's also important to understand the pattern/probability distribution of service times. This distribution might depend on the number of patients in line or the experience of the server. The server may work faster as the line grows or an experienced server may simply work faster than an inexperienced one. We also assume that the patient arrival and service patterns are independent of each other. The following are the key inputs for queuing theory to work well:

- The number of servers which refers to the number of servers available for patients to use simultaneously, so patients can be served from servers from a single queue or multiple queues.
- The capacity of the system refers to the physical limitations of say, a waiting room. The assumption here is that when a waiting room is full, the patient must leave since there's no room for them to sit and wait.
- The queue discipline is a description of the manner in which the patients are served once the queue has formed. The most common disciplines are FCFS, meaning first come first served; LCFS, meaning last come first served; and RSS which stands for random selection service.
- The model notation of the queuing process are described by a series of symbols and slashes as follows: A/B/X/Y/Z with the following definitions of each entity:
- A: Patient Arrival Distribution
- B: Patient Service Distribution
- X: Number of Servers
- Y: System Capacity
- Z: Queue Discipline

Using this model notation, the most common queuing models are M/M/c/infinite/FCFS. The M refers to something called a Markovian process which assumes both the arrival or service rates follow a Poisson Distribution and the time between arrivals and service follows an exponential distribution. The "c" refers to the number of servers. This system assumes that the system capacity is infinite and the queue discipline is FCFS. Have I totally confused you yet? Fear not, this story has a happy ending.

There are two true measures of interest when looking at queuing theory and they are:

- Utilization or how much of the time are the servers busy delivering their service?
- Wait time of how long to patients have to wait to receive the service?

Both of these are always in conflict with each other in that as utilization increases, wait time also increases and conversely, as utilization decreases, wait time also decreases. So one of the uses of queuing theory is making better decision about staffing needs.

The key formula for queuing is ρ = λ / (c * μ)
where ρ is utilization, λ is arrival rate, c is the number of servers and μ is the service rate of the server.

Ok, that's enough for you to absorb in one posting. In my next posting, I'll show you how easy it is to calculate the key factors from this queuing theory formula. I will also try to give you a link to a free Excel file that Cornell University has made available to anyone teaching queuing theory.

Bob Sproull
## No comments:

Post a Comment