One line or multiple lines . . . That is the question!

All of us have been taught that having a system with one line is �better� than having dedicated lines (one per server). Engineers are taught about this in almost every Operations Research course, in almost every University around the world. Indeed, Queuing Theory provides strong support for this �fact�. It can be �demonstrated� that the average wait time for customers in the one line setting is far less than the average wait time for customers in the dedicated multiple lines setting, ceteris paribus. ("all things being equal").

Well, I disagree. The average wait time in both configurations are almost identical. Therefore, when setting aside �customer care� considerations (in particular �fairness�) there is no technical reason to choose one configuration over the other. And I will demonstrate it!

First of all, models traditionally used in Operational Research are based on very unrealistic assumptions, namely Poisson arrival processes and exponentially distributed service times (usually called the M/M/c model, where c stands for the number of servers). During my whole engineering career, I have never fitted an exponential distribution to processing times. The real world does not behave like that!

So, let�s assume for a moment that we do have a system that acts as that beautiful and neat �chalkboard� M/M/c model. Let's use the splitting property of the Poisson distribution to divide the problem in identical M/M/1 sub-models (1 server). The arrival rate for the sub-systems are identical and equivalent to 1/c the arrival rate that the whole system faces. So, using traditional Queuing Theory formulas, it could be demonstrated that the average wait time in the one line setting is indeed far less than the average wait time in each of the sub-systems.

We can verify this using a simple example. Let�s assume we have an M/M/4 system with an average arrival rate of 11.5 customers per hour and average service rate of three customers per hour in each server. Using the Sakasegawa*approximation for M/M/4 we can calculate the average wait time for this one line system as follows:

Now, using the splitting property of the Poisson distribution, we can elegantly assume that the arrival rate for each of the four M/M/1 sub-systems is 2.875 customers per hour (11.5 arrivals multiplied by 0.25, assuming equal probabilities for each of the four Poisson sub-streams). Now using the same Sakasegawa*approximation for M/M/1 (c=1) we get:

 In other words, this stylish and neat theory seems to support the assertion that the average wait time in the one queue system is less than the average wait time in the same system with multiple dedicated queues. Similar kind of �demonstrations� can be found in dozens, maybe hundreds of Operations Research text books around the world. Well, this is too good to be true. Something smells �fishy� here!

To validate the results, I prepared three separate models and simulated the same M/M/4 system in ExtendSim (� Imagine That Inc). The models are:

1. One line,
2. Four lines with one fourth of the arrival rate each (equivalent to four independent M/M/1 sub-systems), and
3. Four lines where customers select the shortest line after arriving to the system (�Intelligent behaviour� model).

The following figure shows the three models and the average wait times after a simulation time of 100,000 hours:

These are the results for the average wait time:

• One line: 1.86 hours.
• Four lines with one fourth of the arrival rate in each of the M/M/1 sub-systems: 7.43 hours.
• Four lines where customers select the shortest line (�Intelligent behaviour�): 2.15 hours.

In spite of its elegance, the neat mathematical assumption of having one 1/c of the arrivals assigned with equal probability to each of the individual lines (the splitting property of the Poisson distribution used in Queuing Theory) virtually never occur in real world situations. When having multiple lines, people and jobs are directed almost always to the shorter line (�Intelligent Behaviour�). So, myth busted!

The statement �The average wait time for customers in the one line setting is far smaller than the average wait time for customers in the dedicated multiple lines setting, ceteris paribus� is not applicable to most real world situations. As proven using computer simulation, there is almost no difference in wait times between the two configurations (1.86 hours for the one line configuration and 2.15 hours for the four lines with intelligent behaviour ). Therefore, the only reason some organizations prefer one line instead of dedicated lines is �fairness�.

If on top of all this, we have to remember that service times never distribute exponential in the real world. Therefore we can affirm that this neat Queuing Theory model is not the best option to deal with real world problems. I hope that after reading this article, professionals of the XXI Century will realize that computer simulation is the right tool to analyze queuing systems in the real world.

In the movie �The Imitation game� the Alan Turin character says �What if only a machine can defeat another machine?� I would like to adapt this question and say: �What if the complexity of the real world can only be analyzed using computer simulation?�

* Whitt, W, 1993, Approximating the GI/G/m queue, Production and Operations Management 2(2): 114-161.

Rene Alvarez, IE, MEng
www.SmartSimulation.ca