| Summary: | In this paper, we consider two single server queueing systems to which customers of two distinct priorities (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula>) arrive according to a Marked Markovian arrival process (MMAP). They are served according to two distinct phase type distributions. The probability of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula> customer to feedback is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> on completion of his service. The feedback (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula>) customers, as well as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> customers, join the low priority queue. Low priority (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula>) customers are taken for service from the head of the line whenever the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula> queue is found to be empty at the service completion epoch. We assume a finite waiting space for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>1</mn></msub></semantics></math></inline-formula> customers and infinite waiting space for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> customers. Two models are discussed in this paper. In model I, we assume that the service of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> customers is according to a non-preemptive service discipline and in model II, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> customers service follow a preemptive policy. No feedback is permitted to customers in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>2</mn></msub></semantics></math></inline-formula> line. In the steady state these two models are compared through numerical experiments which reveal their respective performance characteristics.
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