How can we program Reinforcement learning without transition probability and rewards? I would like to design the optimal task distribution system using Reinforcement learning.
The best advantage of Reinforcement learning compared to traditional Dynamic programming is that it is not needed to know the information of dynamics such as transition probability.
From what I studied for the last days, Reinforcement learning learned that information from experience. 
But what does it exactly mean learning from experience?
For example, using what I want to design, We don't have exact data sets of task demand from service centers, but we know the distributions of task demand for them. 
In this case, every iteration, the next states, and the rewards can be random variables that we don't know exactly what could happen. 
But, because we know the distribution of task demand, we may be able to design following that distribution function in order to decide the next states and the rewards. 
Is it an available framework?
Is it okay that the next states and the rewards (s',r) are different depend on iteration even though the same state and action (s,a)?
In other words, one case can be (s1,a1) -> (s3,r1) but another case can be (s1,a1) -> (s5,r2).
 A: 
Is it an available framework? Is it okay that the next states and the rewards (s',r) are different depend on iteration even though the same state and action (s,a)?

Yes and yes. From what you described it sounds like it can be implemented using Q learning in a system like  , where R is your defined reward.
In detail, you can define your own functions on how transferring from one state to another happens, and also set limits on what actions you can take (e.g. if my states are prices, I can set state transfer function as price dynamics. For actions: for example, if you're optimizing inventory clearance, you can only clear 10 at maximum each time etc). 

But what does it exactly mean learning from experience?

In training your reinforcement system, say you're training a Q table.The more times you train the system, the more experience you have until the optimal solution converges (but of course you'll need to have everything needed to calculate reward, e.g. historical data). Hence in the end you get a set of optimal actions under historical states ready to be used in out of sample testing.
A: Is it an available framework? 

Yes, many people use RLlib and openAIgym in python or Ray in java to implement RL algorithms. I know a little about the python ones so I'll mention those in my answer. What you are describing sounds like a standard algorithm from Sutton and Barto which is implemented in the frameworks above. You may need to develop your own openAIgym environment if your problem is a non-standard problem formulation. The environment is a new class which a specified API (collection of functions you must implement. Here is a link to the doc
https://gym.openai.com/docs/#environments
But what does it exactly mean learning from experience?

The learned policy (mapping from states to actions) is not known a priori but is learned by (and intelligent) trial and error. 
