queue) Baccelli and Makowski give bounds for the average response time and higher moments of this quantity both in the transient and steady state situations. Kemper and Mandjes show that for some parameters these bounds are not tight and show demonstrate an approximation technique. For heterogeneous fork-join queues (fork-join queues with different service times), Alomari and Menasce propose an approximation based on harmonic numbers that can be extended to cover more general cases such as probabilistic fork, open and closed fork-join queues. Subtask dispersion The subtask dispersion, defined to be the range of service times, can be numerically computed and optimal deterministic delays introduced to minimize the range. Stationary distribution In general the stationary distribution of the number of jobs at each queue is intractable. Flatto considered the case of two servers (N=2) and derived the stationary distribution for the number of jobs at each queue via uniformization techniques. Pinotsi and Zazanis show that a product form solution exists when arrivals are deterministic as the queue lengths are then independent D/M/1 queues. Heavy traffic/diffusion approximation When the server is heavily loaded (service rate of the queue is only just larger than arrival rate) the queue length process can be approximated by a reflected Brownian motion which converges to the same stationary distribution as the original queueing process. Under limiting conditions the state space of the synchronisation queues collapses and all queues behave identically. Join queue distribution Once jobs are served, the parts are reassembled at the join queue. Nelson and Tantawi published the distribution of the join queue length in the situation where all servers have the same service rate. Heterogeneous service rates and distribution asymptotic analysis are considered by Li and Zhao. Networks of fork–join queues An approximate formula can be used to calculate the response time distribution for a network of fork–join queues joined in series (one after the other). Split–merge model A related model is the split–merge model, for which analytical results exist. Exact results for the split-merge queue are given by Fiorini and Lipsky. Here on arrival a job is split into N sub-tasks which are serviced in parallel. Only when all the tasks finish servicing and have rejoined can the next job start. This leads to a slower response time on average. Generalized (n,k) fork-join system A generalization of the fork-join queueing system is the fork-join system where the job exits the system when any