Non-Markovian Queueing Systems
The M ∕ G ∕ 1 queueing system (Fig.8.1) is similar to the M ∕ M ∕ 1 queueing system and the only difference is that the service time distribution is no exponential. First we mention some idea, most of which were described in the previous chapter in connec
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Non-Markovian Queueing Systems
8.1 M=G=1 Queueing System The M=G=1 queueing system (Fig. 8.1) is similar to the M=M=1 queueing system and the only difference is that the service time distribution is no exponential. First we mention some idea, most of which were described in the previous chapter in connection with an M=M=1 system.
8.1.1 Description of M=G=1 System Conditions of functioning: 1. At the starting moment 0 the system is empty. For the sake of simplicity we generally assume 0 D 0 (Fig. 8.2). 2. fN.t/; t 0 g describes the number of entering customers; this is a Poisson process with intensity > 0. 3. There is one server functioning without breakdowns; after having served a customer it immediately starts serving the next one. If a customer enters the system and the server is busy, the customer joins the waiting queue. There is no limitation on the queue’s size. 4. The service discipline is FCFS (FIFO). 5. The service times are independent identically distributed (i.i.d.) random variables with distribution function P .Y < x/ D B.x/ and mean E .Y / D B < 1, and they do not depend on the arrival process. The main characteristic of the system is the queue length fL.t/; t 0g, i.e., how many customers are in the system at moment t. Let L.t/ be continuous from the right, i.e., L.t/ D L.t C 0/; t 0. In the case of queueing systems, the basic issues concern the distribution of queue length, whether there exists a limit distribution and how it can be found, the L. Lakatos et al., Introduction to Queueing Systems with Telecommunication Applications, DOI 10.1007/978-1-4614-5317-8 8, © Springer Science+Business Media, LLC 2013
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8 Non-Markovian Queueing Systems Customer arrivals:
Fig. 8.1 M=G=1 system
Poisson process
Service time distribution: B(x)
with rate λ
Fig. 8.2 Number of customers in M=G=1 system
L(t)
s1
s2 Y1
s3 Y2
Y3
s4s5 s6 Y4 Y5 Y6
s7 Y7
4 3 2 1 τ 0 = 0 τ1
t1
τ2 τ 3 t 2 τ 4 τ 5 τ 6
t 3 t 4 t 5 t 6 τ7
τ8
t
average number of customers in the system, etc. In this chapter we will deal with the asymptotic behavior of queue length L.t/ as t ! 1. We introduce the following notations: • 0 CX1 : moment of entry of first customer; Xn : interarrival time between .n1/st and nth customers; • n D 0 C X1 C : : : C Xn .n 1/: moment of entry of nth customer; • Yn : service time of nth customer; • sn ; n 1: starting moment of service of nth customer; • tn ; n 1: moment when nth customer leaves system (service in system is completed at this moment). According to these assumptions, f.Xn; Yn /; n 1g is a sequence of i.i.d. random variables, where the components of vectors are independent, too. Furthermore, the intervals between consecutive arrivals Xn ; n 1, have exponential distribution with parameter , the service times Yn ; n 1, have distribution function B.x/. It is also clear that fn ; n 1g, are moments of jumps of the Poisson process N.t/ (Fig. 8.2).
8.1.2 Main Differences Between M=M=1 and M=G=1 Systems For the M=M=1 system both the interarrival and service ti
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