Async client can only process 2 instances at at time

This is my testing script:

import asyncio
from ollama import AsyncClient

async def run_query_async(model, query, instance_id):
    try:
        client = AsyncClient()
        print(f"\nInstance {instance_id}: Starting query")
        stream = await client.chat(
            model=model,
            messages=[{'role': 'user', 'content': query}],
            stream=True
        )
        async for chunk in stream:
            content = chunk['message']['content']
            if content:
                print(f"Instance {instance_id}: {content}", end='', flush=True)
        print(f"\nInstance {instance_id}: [DONE]")
    except Exception as e:
        print(f"\nInstance {instance_id}: Error - {str(e)}")

async def add_query(model, query, instance_id):
    task = asyncio.create_task(run_query_async(model, query, instance_id))
    print(f"\nAdded new task: Instance {instance_id}")
    return task

async def main():
    model = "phi3"  # Replace with your desired model
    
    queries = [
        "Write a step-by-step guide on how to bake a chocolate cake from scratch.",
        "Develop a python function that solves the following problem, sudoku game",
        "Create a dialogue between two characters that discusses economic crisis",
        "In a forest, there are brave lions living there. Please continue the story."
    ]
    
    # Start with 2 initial queries
    tasks = [
        await add_query(model, queries[0], 1),
        await add_query(model, queries[1], 2),
        await add_query(model, queries[3], 4)
    ]
    
    # Wait for a bit before adding more queries
    await asyncio.sleep(5)
    
    # Add 2 more queries
    tasks.append(await add_query(model, queries[2], 3))
    
    # Wait for all tasks to complete
    await asyncio.gather(*tasks)

if __name__ == "__main__":
    asyncio.run(main())

I have set my environment variable OLLAMA_NUM_PARALLEL=4 so it should be able to handle all instances at the same time. But for some reason it only processes 2 and waits for one to finish to process the next one.

Can it also be that my system is just not able to process more than 2 instances? Or is there something I am missing?

Start with the provided asyncio script and its use of AsyncClient.chat(..., stream=True), then reproduce the behavior with OLLAMA_NUM_PARALLEL=4. Trace whether the client or the configured environment limits concurrent streams; done means all four submitted queries can run concurrently, or the issue documents the actual limiting setting.