ABSTRACT
In “M/G/1” queues, the letter “G” may represent different service mechanisms. “G” may represent the type of random variable for the service-time distribution of the arriving customers. For example service time may depend on waiting time in the queue. Or, servivice time may depend on the number of customers left waiting when the arrival starts service, etc. In general, G is distributed differently from a pure Exp μ random variable. Also, “G” may be a discrete random variable such as a “fixed amount of time” https://www.w3.org/1998/Math/MathML"> D > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math3_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for each arrival. Or, “G” may be a continuous random variable, like a uniform r.v. on an interval https://www.w3.org/1998/Math/MathML"> 0 , a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math3_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , https://www.w3.org/1998/Math/MathML"> a > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math3_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , denoted by “U(0,a) ”, https://www.w3.org/1998/Math/MathML"> a > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math3_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . “G” may be a linear combination of several Exp μ s, e.g., a “generalized hyper-exponential” distribution (see [8]; [60] (2012)). (Note: If G: = Exp μ , the notation used is M λ /M μ /1, as in Chapter 2.
