M/M/1/K Example: Printer Buffer
Let's look at a simple example of an M/M/1/K queuing system. One such example is a printer buffer. These buffers sit between a personal computer and a printer. They accept print data from the computer as fast as the computer can send it, store the data in local memory, and parcel it out to the printer as the printer is able to process it.
For our example, let's make some specific assumptions. Suppose we print an average of 50 documents per 8-hour day. Here's our system's average arrival rate in documents.
To find the average service rate we need to know the printer's speed and the average document size. Suppose our printer can print 4 pages per minute and our documents average 38 pages each. This tells us both the service rate and the traffic intensity.
Since we're doing so much printing, we want to buy a printer buffer to relieve the burden on our computer. When we buy the printer buffer, we have to choose how much memory to get with it. Our choices range from 512 KBytes to 2 MBytes, in 512 KByte increments.
Since the customers in our system are documents, we'll need to convert these buffer sizes from KBytes to documents. To do that, let's assume that each page averages 2000 bytes of data. Our documents average 38 pages, so we can write a simple expression for possible system capacities. Because system capacity includes the server (in our case the printer) as well as the queue (the printer buffer), we also have to include the 512 KBytes of memory already installed in our printer. K is our system capacity, measured in documents; it's a function of the buffer size.
Now we've defined all the parameters for our system. In order to use the M/M/1/K results, we'll have to make two more assumptions. So far, we haven't said anything about the arrival and service processes other than defining their average rates. For an M/M/1/K system, both of these processes must have an exponential distribution. This means that the time between documents must be exponentially distributed, and the document length (which is proportional to the service rate) must also have an exponential distribution. Fortunately, neither assumption is too unreasonable, so we can proceed with the analysis.
A particularly important measure for any M/M/1/K system is its blocking probability. The blocking probability, pK, is the probability that the system is full. Any customer that arrives to find a full system is blocked. That customer has no choice but to leave immediately. In our example, pK is the probability that our printer buffer fills up. When that happens, the buffer can't accept any more documents, and our computer has to wait.
The value of pK comes directly from the equation for state probability pn. You can find that equation in the M/M/1/K summary of results. Here it is for pK as a function of K and intensity.
One of the things we can do with this equation is help us decide how much memory to buy for our print buffer. Suppose we want to be at least 95% sure that we will not overflow the buffer. The probability of not overflowing the buffer is 1 minus the probability of overflowing the buffer. And that probability is the blocking probability, pK.
Let's plot the probability of not overflowing the buffer for various buffer sizes. Since we wish that probability to exceed 95%, we'll also plot a horizontal line at 95.
A quick look at the graph tells us that we need to buy 1 MByte of memory with the print buffer. With that much memory, the chance of overflowing the buffer is just over 4%.
Now that we picked out a printer buffer, let's look a little more closely at the M/M/1/K system we've set up. Notice that the blocking probability, pK, depends on traffic intensity as well as capacity. What would happen if our estimate for a was off? That could happen if we ended up printing more or less documents per day, or if our average document size changed significantly.
Let's consider two possibilities. For the first, suppose we only have to print half as many documents as we expected. For another possibility, let's see what happens if our average document size doubles. We'll graph both of these cases, along with our original case, in the following plot. Note that pK is plotted using a logarithmic scale.
Notice the slopes of the lines. The high intensity system is almost horizontal. That tells us that we can't change its blocking probability much, no matter how big a buffer we buy. The low intensity system, on the other hand, has a very steep slope. (Remember, the y-axis is logarithmic.)
For that system, increasing the buffer size can have a tremendous effect.
Let's use some actual numbers to see this effect more clearly. Suppose we have a high intensity system (because our average document size is double what we assumed earlier.) Nearly half of the time we're waiting for the printer.
Suppose we were to go out and buy a huge printer buffer, one with 64 MBytes of memory.
Well, we would manage to lower the blocking probability some. But it's doubtful that we could justify buying 64 MBytes of memory if it only decreases that probability by a quarter of a per cent!
How about our the low intensity system? In that one our intensity is half because we're only printing half as many documents in a day. Without any printer buffer, our blocking probability is relatively low.
In an 8-hour day, we only have to wait for the printer about 2 minutes.
Now let's add a small printer buffer, one with only 512 KBytes of memory. With this buffer, it takes over 3 weeks for our computer to wait 2 minutes!
We can summarize this behavior this way: For traffic intensities less than 1, capacity has a big effect on performance, but for traffic intensities greater than 1, capacity has little effect.
It's probably not that important for a printer buffer, but with some M/M/1/K queuing systems, we're also interested in the idle probability. That's the probability that the server is idle. We can find the probability from the same equation we used for blocking probability. You can see the equation in the summary of results. Here it is as a function of capacity.
Let's plot this function for the different potential printer buffers.
Does the shape of this curve surprise you? The more capacity we add to the system, the less likely we are to find the server idle. Indeed, that is the case. Here's why. Capacity doesn't affect how much work the server is asked to do. That depends solely on the intensity. What capacity does affect is how much work the server can do. Without a lot of capacity, the system fills up quickly, and potential customers are turned away. In a true M/M/1/K system these customers are lost forever. (Of course, when a real printer buffer fills up, subsequent documents aren't necessarily lost. By using an M/M/1/K model, though, we've assumed that to be the case.) The lower the system capacity, the more customers are turned away. And if a system turns away more customers, it ultimately has less work to do and, consequently, a higher idle probability.
Effective Arrival Rate
Another way to consider this same result is through the effective arrival rate. Since some customers are turned away, the system doesn't really see an arrival rate of l. That's how many customers try to enter. They only succeed if the system isn't full. Here's a way to show this on a queuing system picture.
Notice how the stream of arriving customers divides into two separate streams at the entrance to the queue. The fraction that manage to enter the system is the same as the fraction of the time that the system is not full. We've seen that fraction already; it's equal to 1 - pK. The effective arrival rate, therefore is the product of l and (1 - pK). Here's how we can write it as a function of intensity and capacity.
As you can see from the following graph, the effective arrival rate approaches l as our capacity increases. The units are arrivals per hour.
There are several other performance measures for M/M/1/K systems. They're not particularly important for printer buffers, but they may be for other systems. We'll look briefly at them below. To attach some real numbers, let's pick an actual system capacity based on our initial requirements.
Average Number in System
The average number in the system depends only on the intensity and the capacity. For our printer buffer, that number tells us that our system contains, on average, just under 10 documents.
Average System Time
We can use Little's Law to calculate how long a customer stays in the system. This number tells us how long it takes, on average, for a document to finish printing. In our system that time is about an hour and a half.