It's really going to depend a lot on the particulars. Since we are talking about an arrow wound, I will presume that you are thinking more of a battlefield type scenario as opposed to a hunting accident. To focus a bit more on an area where I have some background, I'll presume European as well: in other parts of the world the approach might be different.

So you happen to find yourself on a European battlefield in the 1400s and take an arrow to the knee, the good news is that you manage to be taken to the rear and are going to receive treatment. What happens next is going to depend quite a bit on your luck and the skill and knowledge of whomever treats you. Unfortunately this is a period in which knowledge of the ligature of vessels was lost, so hopefully this is a simple wound and the arrow is cleanly extracted. Barber-surgeons of the time generally preferred to avoid enlarging the wound (in contrast, modern surgeons will use debridement to remove damaged tissue and help promote healing), so the next step would be to scald the wound with hot oil, use branding iron for hemostasis, then possibly suture the wound closed, perhaps apply a salve, and finally dress the wound. After which you pretty much need to hope that you are lucky and infection doesn't set in.

The salve that may or may not be applied may or may not be useful and there can be some variability here. Some unlikely combinations have proven to be effective, honey has been proven to be a possible antibiotic, and likewise for various herbs that might be used. However, recipes are always tricky and may only be known of within specific regions, and they dependent upon the availability of ingredients. The dressing used may or may not work in your favor as well. A clean cloth would work in your favor, but if they are just working with that they have, then you might end up with something that is contaminated and will lead to infection.

So that is for a simple injury with the arrow cleanly extracted, what about something more complicated? So if you have the bad luck of the femoral artery being severed you likely just bleed out in short order. Likewise for most other scenarios in which there is major vascular damage. Remember that the techniques for the ligature of vessels have been lost, so the ability to stop bleeding is limited to cauterization, pressure, and suturing of the wound itself. Presuming you don't bleed out, then you still have to worry about the possibility of infection. In the event that the arrow was stuck then things really started to come down to the skill of the person treating you and likely your social class as well. In at least one case, an arrow was allowed to fester until the wound was sufficiently full of pus that the arrow could be easily removed (resolving one issue, but now the infection needs to be treated). In other parts of Europe an arrow might be extracted through the use of a cross-bow connected to pliers gripping the arrow, when the crossbow was fired the arrow would be jerked out of the wound, twang! (see Karger et al. 2001 Fig. 4 for an illustration of how this looked).

In summary, your likelihood of serving depended a lot upon the skill of the person attending you, the nature of the wound itself, and what resources were on hand for treatment.

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Karger, B., Sudhues, H., & Brinkmann, B. (2001). Arrow wounds: major stimulus in the history of surgery. World journal of surgery, 25(12), 1550-1555. https://doi.org/10.1007/s00268-001-0168-3