Combinatorial Approach to M / M / 1 Queues Using Hypergeometric Functions

In this paper we use a well known reflection principle for lattice path counting to apply for the analysis of the M/M/1 queuing system. The joint distribution of i arrivals and j departures over a time interval of length t is obtained, when there are k customers in the beginning of the system. The derivation uses the lattice path approach between two points, in two dimensional x-y plane with certain restrictions. Finally a known result of queue length is verified from Satty (1961).


INTRODUCTION
Mohanty and Panny (1990) obtained a transient solution by first discretizing the continuous time model and then representing the same by a lattice path.Sen, Jain and Gupta (1993) used the same approach for solving the M/M/1 model.
In this paper we use the lattice path approach for solving M/M/1 queuing model and obtain an expression for the probability ( , ; ( ) i j k P t ) of i arrivals, j departures in the time interval of length t, under the assumption that the number of customers in the system at time t= 0 is k.We then use this result to verify a known result of Satty (1961).For some related work based on lattice path combinatorics and applications, Case (i) Let r = 0. Fig. 1 In this case the server remains busy in (0,t) and the corresponding lattice path from A(k,0) to B(i+ k, j), as shown in Fig. 1, does not touch the line y = x.Therefore the number of paths from (k,0) to ( k + i, j) not touching the line y=x is given as i j i j j i k In order to find an expression for queue length at time t, we substitute k +i -j = p or i=p + j-k and sum over j (from 0 to ∞ ), we get the same Case (ii) Let r ≠ 0.

Fig. 2
In this case, the server sits idle r times in duration of time interval (0,t) and he has to wait r times for single server.Let T q denote q th waiting time for single server, q=1,2,…,r.In the corresponding lattice path (see Fig. 2) from A(k,0) to B(k+i,j), we

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show this by r contact points with the line y=x and after each contact point the server sits idle for time T q ,q=1,2,…,r for single horizontal step (single arrival).The above Fig. 2 demonstrates the arrival departure process for single server, who sits idle three times (r=3).
The number of paths from (k,0) to (k+i, j) which touches the line y=x , r times is = In this case (

r q t T t T e e dT dT dT i j r
, ; and, therefore, . ) which is the joint distribution for i arrivals and j departures in time interval of length t.

COMPUTATION OF AN EXPRESSION FOR QUEUE LENGTH
To find an expression for queue length putting k+i-j=p (or i=p+j-k ) and summing over j ( from 0 to ∞ ), we get the same It may be noted that within the inner curly brackets, the limits in the first summation are j≥k , and those in the second summation are j≥k +1 .Therefore, putting j-k=s in the first summation and j-k-1=s in the second summation and summing over s ( from 0 to ∞ ), we get the queue length .
)  Since events represented in Case (i) and Case (ii) are mutually exclusive, therefore, the joint distribution of i arrivals and j departures in time interval of length t is given by the sum of equations ( 1) and ( 3) and the expression for queue length at time t is given by the addition of ( 2) and ( 4 ) ) ) as equation (4.24) given inSatty (1961). )