Mathematical Analysis of Queue with Phase Service: An Overview

Table 5

Literature on bulk queues with phase service.

Models	Special features	Authors	Methodology
M/M/1	Bach service, vacation, and additional phase of optional service	Kalyanaraman and Murugan [155]	Probability generating function
		Murugan and Kalyanaraman [156], Sharma [157]
M/G/1	Quorum, bulk arrival, optional reservice	Tadj and Ke [158]	Probability generating function, supplementary variable
	Threshold policy, bulk service, and second optional service	Haridass and Arumuganathan [159]	Probability generating function
/G/1	Bulk arrival, startup, additional optional service	Ke [160]	Probability generating function, Supplementary variable
	server breakdown, retrial, batch arrival, Bernoulli admission mechanism, and two-phase of service	Choudhury and Deka [161]
	Retrial queue, bulk arrival, second optional service, Bernoulli vacation, and unreliable server	Jain et al. [121]	Probability generating function, Supplementary variable, Embedded Markov chain
BMAP/G/1	Batch arrival, retrial, and phase of service	Kim et al. [162]	Matrix method
MAP/PH/1	Batch arrival, -policy, phase service	Kim et al. [163]
	Batch arrival, batch service, -policy, and phase service	Lenin et al. [164]	Newton approximation technique