ABSTRACT
In the standard M/M/c queue customers arrive to the system in a single Poisson stream with rate https://www.w3.org/1998/Math/MathML"> λ > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math4_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and form a single waiting line in front of c (c https://www.w3.org/1998/Math/MathML"> ≥ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math4_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ) independent servers in a parallel arrangement (as opposed to series arrangement; see Figure 4.1). Customers begin service by one of the c servers in order of arrival due to the single line before service. If a customer arrives when some or all of the c servers are idle, the customer begins service immediately by one of the idle servers, with equal probability on the set of idle servers. In the standard M/M/c queue, each server serves a customer at the same exponential rate https://www.w3.org/1998/Math/MathML"> μ > 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429297601/14442e7e-1dcb-40d5-bc26-b3039a19098c/content/math4_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> independently of the other servers. In the standard M/M/c queue customers complete service and depart from the system in random order due to random service times and the parallel arrangement of the c servers. In the standard M λ /M μ /c queue we will derive the stationary probability distributions of key random variables. (Note: For a more general LC analysis of transient (time-dependent) probability distributions of key random variables in M/M/c queues, (see p. 188ff in Chapter 4 in [22] (2017), [12] (1988), and [14] (1988).)
