There are loads of factors, so it is hard to guess, there have even been cases where the starter worked but just didn't look like it had done much. But FWIW perhaps the main concerns are the health of the yeast you collected (e.g. yeast from a high gravity or otherwise stressful brew won't be as good as those from a normal gravity one) and in particular how well and how long they were stored for.
Assuming the slurry is healthy then it is okay to re-pitch direct if there is a lot and it is quite fresh, but it really does start to lose viability quite quickly. If you want to store it for a couple of weeks then keep it under some of the beer that it made (i.e. don't wash it and then keep it in water), and keep it cool in the fridge but not frozen. The
Mr Malty calculator (free if used from a desktop computer) has a slurry tab that will help you work out what kind of amounts of slurry might be needed at what age, it is only a rough guide but better than completely guessing. Make sure to select 'lager' yeast and the other parameters for your brew.
Once viablity gets low (or typically beyond a couple/few weeks IMO), then ideally a starter would be made. The yeast shouldn't be dead (unless something bad happened) but will decrease in number of viable cells as time goes on, so will take progressively more effort or time to grow up in number again. As it gets older still (say months) a stepped starter would be better than a single one, to grow the numbers up in stages.
There are various ways to collect and store yeast, and other methods entirely such as splitting the fresh yeast rather than re-using/harvesting them. But most methods (except directly re-pitching slurry) will involve starters and time. If economy is the key objective then saving boiled wort from brew-day rather than buying extract for the starters can help with that.