Threading

This code should meet your requirements described by pseudo code:

class Program
{
    static object _workLockObject = new object();
    static volatile int _currentPriority = 0;

    static void Main()
    {
        Task t1 = new Task(() => TryDoWork(1));
        Task t2 = new Task(() => TryDoWork(2));
        Task t3 = new Task(() => TryDoWork(3));

        t1.Start();
        Thread.Sleep(100);
        t2.Start();
        Thread.Sleep(100);
        t3.Start();
        Console.ReadKey();
    }
    public static void TryDoWork(params int[] initialConditions)
    {
        var priotity = Interlocked.Increment(ref _currentPriority);
        while (!Monitor.TryEnter(_workLockObject))// starting to wait when DoWork is available
        {
            if (priotity != _currentPriority) // if the thread has stale parameters
            {
                Console.WriteLine("DoWork skipped " + initialConditions[0]);
                return;// skipping Dowork
            }
            Thread.Sleep(300);// Change the interval according to your needs
        }
        try // beginning of critical section
        {
            if (priotity == _currentPriority) // if the thread has the newest parameters
                DoWork(initialConditions);
        }
        finally
        {
            Monitor.Exit(_workLockObject); // end of critical section
        }
    }
    public static void DoWork(params int[] initialConditions)
    {
        Console.WriteLine("DoWork started " + initialConditions[0]);
        Thread.Sleep(5000);
        Console.WriteLine("DoWork ended " + initialConditions[0]);
    }
}

Interlocked.Increment guarantees that every thread has its own priority and the latest thread has a priority that allows to execute DoWork when it is available.