Tolls, Toll Booths, Queues and Queuing Theory

I moved to Gurgaon about 6 months back and have driven on the Delhi-Gurgaon Expressway a number of times. I am not going to even begin sharing my views and opinions on the safety, discipline and road behaviour of the users of this Expressway. That is not the purpose of this blog. I am here to discuss more mundane and boring stuff!!

The amount of news and criticism that this Expressway generates is amazing. Of course most of it is just simple “zero IQ” ravings and rantings  about long queues at “Asia’s largest Toll Plaza”.

For me the long queues and waiting times are a result of the design of the toll plaza and application of my Operations Research -101 lessons.

The Toll Plaza at km 24 caters to a stunning 1,90,000 vehicles per day in both directions. That makes it 95,000 vehicles in each direction [I know I am making some assumptions here].

Making a very general assumption that this traffic is evenly distributed throughout the day [I can hear many of you saying…what the hell…but just hold on] 4.12 cars will arrive per minute at each of the toll booths and assuming a very quick service rate of 4.5 cars per minute, the average queue would be 6 vehicles and waiting time would be 1.5 minutes. Not bad at all!

But this was an idealistic situation which maybe happens over weekends or holidays.

Here is what happens in reality.

About  25% of the traffic arrives between 9am and 12pm noon in one direction [this seems to be a reasonable assumption] and between 6pm and 9pm in the reverse direction. That is when chaos occurs and the system breaks down. Vehicles will now arrive at 8.25 per minute, but will be served at 4.5 per minute leading to an “infinite queue”.  To cater to the peak traffic, 30 booths would be required.

What are the solutions or options available to tackle the peak traffic? Have Tandem toll-booths. This typically increases throughput by 25%.

Encourage more vehicles to get TAGs.  TAGs enable a faster thorughput of vehicles. For this, TAG only booths should be strictly enforced. What happens now is that Cash vehicles [vehicles which do not have a TAG, but wish to pay cash] also come on the TAG booth, thereby slowing the traffic  and throughput of the vehicles who have a TAG.

Have the toll amount in easy coinage. The toll is Rs 21, which always involves giving change.

The reason for this post is to reiterate the importance of scientific and logical planing of logistics infrastructure. Many times we do not plan well while developing warehouses, container terminals, logistics parks or creating infrastructure within our plants. The next time you see a queue at an airport, cinema, railway station or when you hear the message “all our customer service executives are busy, your call is important to us, please be on the line, we will attend to you shortly” you know that someone has goofed up by not paying heed to Queueing Theory.

To end this post, I quote what our professor used to say, “you know the the only way to beat queuing theory is to be 1st in the queue, else you will always be subjected to the laws of queueing theory!”

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