If it was requested early enough, like the day before, then no, since you then had time to adjust your schedule and not lose business on account of it.

I know what traffic can be like here in DC, but that is just me.  I also do not book my appointments so tightly that 15 minutes would screw up my schedule.

but if you did that to me if I gave you a heads up I would be late, it would be the last time I ever saw you.

This is one of those things that yes, you are within your rights to do so, but it's lousy customer relations and not likely to do your rep much good in the future

... it depends on the level of customer service you intend to provide. Since he is a new client, and you wish to keep his business, you should obviously cut him some slack.  

If you made a 6:00 pm dinner reservations, called ahead to let them know you were running 15 minutes late, how would you feel 1) if they gave away your table or 2) said 'no problem, we will hold it for you.'  In one case you will never go back. In the other, you think they walk on water. Good customer service pays off many times over.  

I may be different than others, but I pay different rates based on the level of customer service I get. The ladies that understand this concept make lots of money from guys who have deep pockets.

Not that I know what is exactly transpiring with your daily agenda, but he was considerate enough to call.  Now how did you want to broach the subject of compensation for time is another question.   Did you want to charge him a mere fraction of your hour? Or did you want to modify your website to discuss this with respect to compensation for time and company?  Also another question is he a regular or is this the first time with him as well?  I can imagine that you do a lot during the day and have a myriad of responsibilities.  Perhaps you can truncate this session and then offer to give him a free fifteen minute or 30 if he pays for the hour next time.  That way you can accomplish your immediate missions while giving him his time that he missed.  That way you don't totally alienate a client and you have a chance of a return visit. It also helps if he is a good reviewer because now he isn't slamming you in print and in your wallet as well.  Just a couple of suggested solutions.

if you were able to grant me my full time under those circumstances. However, I would not presume that you could nor would I expect to do anything other than compensate you for the full time agreed.

This has happened to me more than once. Sometimes, we start the clock when I arrive (eternally grateful to those providers who are able to do so) and sometimes we start the clock when I should have arrived.  

As in all aspects of life that don't go perfectly between two people, communication, generosity and the like are the key to handling it easily and without resent on either side

It's up to prostitute holding him accountable for his tardiness on paid appointment..

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it is just a good business practice and guys will repeat hopefully if you are good to them and fair...

if you get into the session and things are going well and you're enjoying yourself and you decide you want his return business, then extend the time by the 15 minutes.

if things are rocky and he's a jerk, you can let him know politely, "i'm sorry, but you were late and i have other appointments and errands i have to get to.  i regret that we will have to end on the scheduled time."

however as others have pointed out, he was kind enough to let you know ahead that he would be late.  if it's a big city with  traffic, that only makes sense.

You can be nice but it really depends on your schedule and the interaction and the overall situation. What time, for instance. In my area, during certain times of the day traffic is really unpredicable -- a trip can take 15 minutes or 40 minutes.  

Still I've always though the providers needs to schedule property and I need to plan on being on time -- if it's a bad time of day I've set the appointment for and I don't leave early enough that is on me. I pay the full period and am ready to leave per the original schedule.

In generally I'd say unless you have other commitments adjusting to take into consideration the late start is a great way of getting an better review and following the "under promise, over deliver" model of customer service.