Threadsafe FIFO Queue/Buffer

I need to implement a sort of task buffer. Basic requirements are:

• Process tasks in a single background thread
• Receive tasks from multiple threads
• Process ALL received tasks i.e. make sure buffer is drained of buffered tasks after a stop signal is received
• Order of tasks received per thread must be maintained

I was thinking of implementing it using a Queue like below. Would appreciate feedback on the implementation. Are there any other brighter ideas to implement such a thing?

public class TestBuffer
{
    private readonly object queueLock = new object();
    private Queue<Task> queue = new Queue<Task>();
    private bool running = false;

    public TestBuffer()
    {
    }

    public void start()
    {
        Thread t = new Thread(new ThreadStart(run));
        t.Start();
    }

    private void run()
    {
        running = true;

        bool run = true;
        while(run)
        {
            Task task = null;
            // Lock queue before doing anything
            lock (queueLock)
            {
                // If the queue is currently empty and it is still running
                // we need to wait until we're told something changed
                if (queue.Count == 0 && running)
                {
                    Monitor.Wait(queueLock);
                }

                // Check there is something in the queue
                // Note - there might not be anything in the queue if we were waiting for something to change and the queue was stopped
                if (queue.Count > 0)
                {
                    task = queue.Dequeue();
                }
            }

            // If something was dequeued, handle it
            if (task != null)
            {
                handle(task);
            }

            // Lock the queue again and check whether we need to run again
            // Note - Make sure we drain the queue even if we