Abstract
In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing ongoing service batch, irrespective of how long service was provided to that batch. A queuing system with a different kind of accessibility that relates to a real situation is studied in the paper. Consider a single server queuing system in which the service process comprises of k stages. Customers can enter the system for service from a node at the beginning of any of these stages (provided the pre-determined maximum service batch size is not reached) but cannot leave the system after completion of service in any of the intermediate stages. The customer arrivals to the first node occur according to a Markovian Arrival Process (MAP). An infinite waiting room is provided at this node. At all other nodes, with finite waiting rooms (waiting capacity cj,2≤j≤k), customer arrivals occur according to distinct Poisson processes with rates λj,2≤j≤k. The service is provided according to a general bulk service rule, i.e., the service process is initiated only if at least a customers are present in the queue at node 1 and the maximum service batch size is b. Customers can join for service from any of the subsequent nodes, provided the number undergoing service is less than b. The service time distribution in each phase is exponential with service rate μjm, which depends on the service stage j,1≤j≤k, and the size of the batch m,a≤m≤b. The behavior of the system in steady-state is analyzed and some important system characteristics are derived. A numerical example is presented to illustrate the applicability of the results obtained.
Highlights
A detailed literature survey of bulk service queueing systems can be found in [1,2]
In most of the works, customer service is provided in batches of varying sizes with minimum batch size a and maximum batch size b- called general bulk service (GBS) rule
The assumption made in that admission strategy is that customers who join during an ongoing service batch will not increase the total service time of the batch
Summary
A detailed literature survey of bulk service queueing systems can be found in [1,2]. In most of the works, customer service is provided in batches of varying sizes with minimum batch size a and maximum batch size b- called general bulk service (GBS) rule (introduced by Neuts [3]). In [11,12], a finite (infinite buffer) bulk queuing system with renewal input and exponential service provided either singly or in batches, depending on the number of customers present in the system, is studied (the difference being that, in [12], accessibility to an ongoing service is restricted to a threshold value d). The queuing system considered in the present paper overcomes these shortcomings as service is assumed to be provided in stages, and customers can access an ongoing service batch from the beginning of these stages, depending on their requirement. This queuing model is immediately seen to be applied in elevators and transport systems. Even with a Marked Markovian arrival process (MMAP) assumption for arrivals to all nodes, there is a dimensionality problem
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