INTRODUCTION AND DESCRIPTION OF THE PROBLEM WITH ITS SOLVING
Luczak and McDiarmid (2006) has done study in the supermarket
model and considered that the distribution of the arrival of customers follows
the Poisson distribution, the model M^{x}/G/1 was in working by Koole
et al. (2005) has also Poisson distribution for arrival of customers
but in the study of Sadowsky and Szpankowski (1992)
was in Multiserver G/G/c queue where the distribution of arrival customers
is general distribution. Tadj and Hamdi (2001) performed
maximum entropy solution to a quorum queueing system. The M/G/1 FB Queue was
performed by Nuyens (2004) and carried out the Maximum
Queue Length for HeavyTailed Service Times. Artalejo et
al. (2007) carried out algorithmic analysis of the maximum queue length
in a busy period for the M/M/c retrial queue. In this study added a new case
involving distribution of interarrival times of customers follows the standard
normal distribution. The aim to get the formula of the probability of the maximum
queue length in the system at this new case.
Let q_{n} be the number of customers found in the system immediately prior to the arrival of the n th customer C_{n}. The sequence {q_{n}} forms a discrete state Markov chain. Also, let X_{n+1} be the number of customers served between C_{n} and C_{n+1}. Thus q_{n+1} = q_{n+1}X_{n+1}, where X_{n+1}≤q_{n}. Since the service distribution is exponential with rate μ, It follows that:
where, y is the time between consecutive arrivals, Let:
where, γ_{k} (y) is the number of customers in the queue at time y, under the condition that at the time 0 the customer enters the system which already contains k customers. Thus at the time 0 there are k customers in the queue and one is being serviced. It is evident that:
where: A_{1 }is the event that a customer arrived at time y, and
k = 0, 1, ..., n given that y varies from zero to t. A_{2} is the event
that a customer arrived at time y,
and given that y varies from zero to t. A_{3} is the event that no customer
arrived during the time interval [0,t] ≡ the event that the interarrival
time is greater than t. It is clear that:
But:
And:
Let: y = r sin θ, z = r cos θ,
Then: dy.dz = rdr.dθ, where,
and it
follows that:
Hence:
Therefore,
But:
And:
Hence:
Therefore:
where,
Similarly:
And:
Then:
Application: Suppose that: Arrival rate = λ = 1, Service rate = μ = 2, t = time in min = 1, 2, 3 n = maximum queue length = 12, 13, 14, 15, 16.
Applying the last formula for the probability of a maximum length of a queue in the system during the period of time t and using mathematic program on the Eq. 1 as follows:
If t = 1, then:
Similarly, P_{14} (t = 1) = P_{15 }(t = 1) = P_{16 }(t
= 1) = 0.60247≈0.6025.
On the other hand, when t = 2, then we can get:

Fig. 1: 
Maximum queue lengths and its probabilities 
From Fig. 1 we can deduce the following information:
• 
Probabilities for a maximum length of a queue are approximately
equality and equal to a fixed amount at t = 1 min. means that as long as
the time a small odds are the maximum length of a queue is approximately
equal and equal to a fixed amount and the relationship between these probabilities
and the maximum length of the queue to be a linear relation as the following: 
• 
Similarly, in the case of t = 2 min. the linear relation is
written as follows: 
But, when t≥3 the values of the probabilities of the maximum queue length will be greater than one, so this is contradictory that because:
Similarly, The values of P_{13 }(t = 3), P_{14 }(t = 3), P_{15} (t = 3), P_{16 }(t = 3) will greater than 1. Thus, we can estimate the limits of time appropriate to the values of the maximum lengths of queues in the system or try to change the service rate or the rate of arrival or the values of the maximum length of a queue as long as we have been given that and reiterate once again enter the values in the formula that we have to get the desired results.