Hello, all. I'm a first time DM, and I decided to write my own campaign with an "open world" philosophy. Essentially I attempt to railroad as little as possible, preferably not at all. There is a setting that is already established, with key players, a mission to complete (save the world, of course), and some goals that must be achieved.

But the concept is the players write the story about how the accomplish it. I don't have a pre-written module that sequences the events from start to finish. I essentially write the details week to week, inserting consequences of decisions made from the previous week and perhaps even a few sessions earlier. I'll throw a problem at them, and I'll have a loose plan for various courses of action that I think they may choose. I LOVE it when they come up with creative solutions that I had not though of.

Last session this caused a really long delay as the players debated courses of action based on the information and circumstances that I threw at them. Has anyone else experienced this? Is it something I need to address, or is it just something you have to expect from time to time?

I play very similarly.

There's nothing wrong having a session where "nothing" but deliberation happens.  Eventually you'll probably have sessions where nothing but recapitulation happens.  I'd say you're on the right track if the players are invested in the deliberation and no one's getting frustrated by it.  I don't think there's any D&D Authority with a stopwatch or role playing metronome telling anyone's game their playing is off the clock or tempo standard.  This play style is only problematic if the people engaged in it find it frustrating.  (As the DM, you've kinda non-structured it this way, so you. don't get to complain).    

The "open world" philosophy is basically whatever RPG would offer so I think the point here for you to see it as how much or how little you put limitations on accomplishing a goal. So, the players can built they're own stories, but you still need to focus on how to built the narrative over whatever they are guiding it so you should probabbly take some notes on possible results on the course of their actions (as much as the possible choices they can make).

I'm also DMing a table in that kind of style (improv sessions around the clues I have from a previous one), but I do DM for a bit of time and I would say that this PbA is pretty common when you DM the game in that way.

The players would still need you to ride the narrative and when they feel free enough to choose theyr own destinies they may feel a lack of knowledge on doing it. They don't know exactlly how that choices will change the narrative and that feeling of freedom becomes a feeling of having no control at all. It also can "railroad" them to act awkardly, experimenting the limits of the narrative, and probablilly leding them out of their missions or turning it into a boring sandbox. (Edit: not that I don't like sandboxes, I meant that it can become boring for some becouse of losing porpouse.)

So, what I try to do with my tables in that style of play is to not stick on that "open world philosophy" to much to not intervene when I see need to. Whenever you feel that players are getting loose on the fun of the game, or they seem unable to progress into the narrative, railroad them a bit. Give them clues on wheter they can go from there, so they will feel confident on doing it. Also, they spect you on doing it, thats your role on the narrative, and they shouldn't feel as they did not have control over the narrative, but would feel as they have something factual over which they can control.

From my experiance even the most open world campaign needs elements of railroading to actually give the sense to the players that they are achieving something. The main issue comes form the fact that the players can never know all the information or experiance that the characters would have after 20+ years of living in this world. I have also found that what can sometimes seem the most obvious plot thread to a DM goes right over players heads and they get lost as to what they should do. 

In the current adventure the party are in a town, they have been getting to know the towns folk, done a few jobs for the town to earn some needed coin and slowly stumbled on a wider deeper plot involving an Aboleth and a wizard working together to enslave the members of the town, there are also 3 changelings involved in the event working for the Wizard (changling replaces town member, wizard transports them to aboleth, aboleth enslaves and wizard returns them). This has meant day to day I have had to keep track of every NPC that is on the list to be enslaved and work out are the party interacting with pre enslaved, changling, or post enslaved. last session the party ran out of things to do and although I had left 3 plotlines dangling they went off and wandered aimlessly round town asking the wrong questions to the wrong people, going to the wrong places and missing out on some key bits of info. As the session approached an end I therefore manufactured a situation to get them back onto a thread telling the party they where walking close to the brothel one of the characters was raised in. As expected the Minotaur Barbarian asked to go have a look (some great RP as he asked the party member if she could get him a discount for time with her mum) and the party headed that way. Her mother had been taken and replaced by a changling the day before.

After some clever roleplay the character caught the changling out and the party found ways to force it into it's normal form and got it to answer questions. The only thing I did was dangle the fact that the party, as they walked through town, had found themselves a few streets away from her mothers. Everything from that moment was open world, the players may not have gone, and when they got there they could have resolved the changling in many ways. Including not picking up on the mistakes the changling made. Next session they had a changling and the information it had provided them to deal with, Changling is cooperating because it wants to live and is not a threat in terms of a fight. I will let the session play out I know the limits of what the changling can do, who it has studied and can impersonate in the town if it can use that as a chance to escape. Or they may thwart it's efforts and be able to use it to convince the few people they know are not under mind control of what is going on (one of whom is the wizard which will open another raft of opportunities). 

So I think having a session like this, where it seems not a lot happens, now and again is fine, but be aware and ready to give your players a nudge in the right direction. Also listen to your players away from the table, if they are talking about having no idea what to do next session it is a big hint to you to create a situation next time that gives them direction. I try and avoid anything as obvious as, someone comes in and tells you all this info. Instead I will have things set up. That visit to her mother, if it hadn't clicked with the party I had 2 other random encounters set up ready. 1 would be children trying to pick pocket them, the children would then explain that mummy and daddy have not been looking after them recently. This would have led to them finding 2 enslaved individuals who where low down the totem pole and had almost been forgotten about. There was also another social encounter someone would have been waiting for them at the inn with a potential job, this would have led to another more obvious clue as to who in town was behind the changlings and mind control and in a roundabout way would have led the character back to her mother, potentially after she had been enslaved. 

Another thing I do is that out of game debate and discussion, I will tell the players that in game they have spent 2-3 hours talking through the options, making real time equate to game time. In my head I roughly transcribe every 10-15 mins of out of game discussion to equal about an hour in game and, if they are in an inn, I will then also tell them they have spent a few silver on drink or food while they have been talking. If there is a time element to what they are discussing i will let them know the clock is ticking. It is also good to learn to spot when the party are talking round in circles, at this point I usually step in and just review the main points they have discussed and ask them for a decision on what they want to do next. Or, as in the case above, if it is clear all they are doing is saying they don't know what to do, I will present them with something in game to resolve or deal with and move them on. My rule of thumb is to always end a session with the party feeling they have achieved something no matter how small. 

If your players are talking that much about all the stuff that's happened in your open world campaign, then congratulations, you've done something very right. 

However, if you feel like your players have arrived at a reasonable plan/conclusion and further discussion is more obfuscating the issue than anything, maybe throw an encounter at them to change their mental gears. Suddenly orcs attack as they march along the road, suddenly assassins break into the royal court, the players get word that a crazed beast is tearing up the market, something like this. 

As always, Matt Coville talks about using encounters to combat analysis paralysis in this video: https://youtu.be/31IAzJO-BEA