# Queuing theory: calculate probability of an medical case

1. Oct 22, 2011

### omaiaa0p

queuing theory: calculate probability of an urgent medical case

Hello everyone,

I am trying to learn about queuing theory. I have read about it and would like to understand how to solve the following:

If I run a hospital with patients coming in the emergency room, what is the average time a patient waits and what is the probability of a traumatic patient coming in the room that he must be served before the others and how will that affect the waiting time.

I beleive I can apply little's law..

avg number of people being served = arrival rate * serving time

but I am a little confused, how can I merge the probability part along with the queuing theory part, is there some formula I can find to build on this.

So far, I have computed the average number of patients arriving per day (lambda), and the service rate (mu)..
which, according to this formula

server utilization = row = lambda / mu. Does this help?

I need to know if Im on the right track..

Thanks

2. Oct 24, 2011

### bpet

Re: queuing theory: calculate probability of an urgent medical case

A good way to check is by comparing against randomized simulations; and if you're keen, test the sensitivity to the stationarity assumptions behind Little's formula, e.g. with peak/off-peak periods.

3. Oct 24, 2011

### omaiaa0p

Re: queuing theory: calculate probability of an urgent medical case

Thank you for your reply.

I have done some more reading and research. It seems that I can find the probability of a waiting time by using the Erlang C function, I can also find the average waiting time by an excel function. However, I still cant seem to get the emergency part.

If a non-emergency patient is P(A) does that mean that the emergency case would be 1-P(A)? I also have not understood your reply very clearly, off peaks and on-peaks would be too accurate and I am assuming this based on a random Poisson distribution.

Thanks

4. Oct 25, 2011

### chiro

Re: queuing theory: calculate probability of an urgent medical case

Hey omiaa0p and welcome to the forums.

What kind of queueing system do you have? Is it a simple one where you have a static service rate and "queue" (line-up) rate or is it more general and complicated?

5. Oct 25, 2011

### omaiaa0p

Re: queuing theory: calculate probability of an urgent medical case

Thank you for your reply,

I am basing my question on an M/M/C system with c servers and an infinite number of patients..

I would like to be able to graph the Poisson distribution of this as well. I have calculated lambda but I dont really understand the x = 0,1,2,3.. part, does this mean the interval from 0-1, then 1-2 or does this mean per event occuring, I would like to make sure of my understanding.

Thanks

6. Oct 25, 2011

### omaiaa0p

Re: queuing theory: calculate probability of an urgent medical case

BTW, my system is in equilibrium, so Little's law is applicable..

Thanks

7. Oct 25, 2011

### SW VandeCarr

Re: queuing theory: calculate probability of an urgent medical case

The values of x are the number of events occurring in some fixed time interval, typically one hour in hospital emergency wards. A calculation for a system in equilibrium is based on a rate parameter$\lambda$. It's reciprocal is the mean time to an event. So a rate of 5 events per hour means an expected waiting time of 1/5 hr or 12 minutes between events. $\lambda$ is the only parameter necessary to specify a Poisson distribution which assigns a probability for x over a discrete range of $0 \leq x$

However when one considers service time in addition to waiting time to estimate queue lengths, Little's formula $L = \lambda S - 1$ is used, where L is the queue length and S is the service time. If S is constant, you can use the Poisson distribution to estimate the probabilities for different values of x. However, Little's formula is independent of the probability distribution.

In any calculation, be sure to use the same unit for all times. So for x events per hour, express a service time of 15 minutes as 0.25 so that L=0.25(x)

Edit: Note that this version of Little's formula gives a zero value if the service time exactly matches waiting time. Negative or fractional values simply indicate the efficiency of the system, so a queue length of 3/4 would mean an intermittent queue of 1 for three of four arrivals while a negative value indicates no queue and spare capacity.

Last edited: Oct 25, 2011
8. Oct 25, 2011

### omaiaa0p

Re: queuing theory: calculate probability of an urgent medical case

Thank you for your reply.

When you have calculated 1/5 = 12 minutes of time between each event, does that mean that mu = 12, average time waiting in the system?

How will I be able to use the Poisson distribution to calculate the probabilities of traumatic cases and how will that affect the waiting time, would that be using the Erlang b formula.

No one has yet answered if P(A) was the probability of regular cases, then 1-P(A) is the probability of a traumatic case arriving, is this accurate.

Thank you for your clear explanation

9. Oct 26, 2011

### SW VandeCarr

Re: queuing theory: calculate probability of an urgent medical case

Yes

Good question. I've never set up a combined situation were you have two service times within a single framework. My suggestion is to use your data to modify S to accommodate traumatic cases, So if on average, every fifth arrival is a major trauma and the average service time for these is 30 min (in the EW), then your mean service time is (15(4)+30)/5=90/5= 18min= 0.3 hr. As a practical matter, trauma centers triage major trauma cases into different wards and keep separate statistics. For regular hospitals, major trauma is rare and can be accounted for by the way I suggested.

Erlang's formula for arrival times and service times is mathematically the same as Little"s. Erlang defined arrival times a little differently than the way that term is used in your problem.

.

Yes, assuming there are no other classifications and the total probability of a case is unity, but I'm not sure how you would use that.

Last edited: Oct 26, 2011
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