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This paper deals with the study of multi-server queueing model in a fuzzy environment with imposition of reneging of customers. Entry of the customers in the system is assumed to be Poisson process and exponential service time distribution under first - come-first-served basis. Specific of this investigation is to derive the various fuzzy performance measures such as fuzzy queue length, fuzzy waiting time in queue, fuzzy response time and fuzzy optimal number of servers in explicit form for the finite capacity multi-server queueing system by using recursive method. For the validity of the model we have obtained the numerical illustrations in tabular form which shows that fuzzy-queue can be more realistic than crisp queue.

Since the birth of queueing theory concept in 1909 by the contribution of A. K. Erlang [

Several researchers have contributed to the study of finite capacity fuzzy queueing models. It is worth noting to mention some of the contributions. Pardo and Fuente [

To the best of our knowledge, very rare literatures can be found in the optimization of fuzzy queueing systems so we are motivated to report some of the works done on the line. Lin and Ke [

In real world, many queueing situations arise in which there may be a tendency of customers to be discouraged by a long queue. Consequently, the customers either decide not to join the queue (balking) or depart after joining the queue without getting the service due to impatience (reneging). Queueing systems with balking and reneging have been studied extensively due to their wide applicability in many areas such as communication systems, production and inventory systems, air defense systems, machine repairing systems, ambulance service. An M/M/1 queue with impatient (balking and reneging) customers was first proposed by Haight [

In this paper, we develop the mathematical model of optimization of multi-server finite capacity Markovian queueing model in the fuzzy-environment under the reneging behavior of the customers. The novelty of our model is that it deals with finite-capacity multi-server queueing system by embedding it into fuzzy-concepts. In the problems with maintenance and inventory which have a large number of states and exact information of a particular state is difficult to know and also, automatic machining systems have been designed in fuzzy-concepts that motivated us to study the model under investigation. This model may be of the first of this kind in which optimization of model under fuzzy environment with imposition of reneging of customers has been taken under study and major objective of this work is to determine the various fuzzy performance measures such as fuzzy queue length, fuzzy waiting time in queue, fuzzy response time and fuzzy optimal number of servers.

For our model we have used the following notations:

λ: Mean arrival rate

μ: Mean service rate

γ: Mean percentage rate of change

N: System capacity of queueing model

s: Number of servers

r: Mean reneging rate

P_{n}: Probability that there are n number of units in the system

The state-transition-rate diagram for our model is shown in

With the help of above transition diagram, the steady-state equations are:

μ P 1 − λ ( 1 + γ ) P 0 = 0 (1)

λ ( 1 + γ ) P n − 1 + ( 1 + n ) μ P n + 1 − ( n μ + λ ( 1 + γ ) ) P n = 0 , 1 ≤ n < s (2)

λ ( 1 + γ ) P n − 1 + { s μ + ( n + 1 − s ) r } P n + 1 − { λ ( 1 + γ ) + s μ + ( n − s ) r } P n = 0 , s ≤ n < N − 1 (3)

λ ( 1 + γ ) P N − 1 − { s μ + ( n − s ) r } P N = 0 (4)

Solving Equations (1) to (4) recursively, we have

P n = { 1 n ! ( λ ( 1 + γ ) μ ) n P 0 ; 1 ≤ n ≤ s 1 s ! ( λ ( 1 + γ ) μ ) s ∏ i = s + 1 n λ ( 1 + γ ) s μ + ( i − s ) r P 0 ; s < n ≤ N − 1 1 s ! ( λ ( 1 + γ ) μ ) s ∏ n = s + 1 N λ ( 1 + γ ) s μ + ( n − s ) r P 0 ; n = N (5)

With normalizing condition

∑ n = 0 N P n = 1

We obtain the probability that the system is empty which is

P 0 = [ ∑ n = 0 s 1 n ! ( λ ( 1 + γ ) μ ) n + ∑ n = s + 1 N − 1 { 1 s ! ( λ ( 1 + γ ) μ ) s ∏ i = s + 1 n ( λ ( 1 + γ ) s μ + ( i − s ) r ) } + 1 s ! ( λ ( 1 + γ ) μ ) s ∏ n = s + 1 N λ ( 1 + γ ) s μ + ( n − s ) r ] − 1 (6)

Also, the probability that the system is full is given by

P N = 1 s ! ( λ ( 1 + γ ) μ ) s ∏ n = s + 1 N λ ( 1 + γ ) s μ + ( n − s ) r P 0 (7)

Other performance measures are

1) Expected number of idle servers is given by

E ( I ) = ∑ n = 0 s − 1 ( s − n ) P n

2) Expected number of busy servers is given by

E ( B ) = s – E ( I )

3) Probability that the servers remain busy is

P ( B ) = ∑ n = s N P n

4) Average rate of reneging is

R r = ∑ n = s N ( n − s ) r P n

5) Expected number of customers in the system is

L s = ∑ n = 0 N n P n = P 0 [ ∑ n = 1 s 1 ( n − 1 ) ! ( λ ( 1 + γ ) μ ) n + ∑ n = s + 1 N − 1 n s ! ( λ ( 1 + γ ) μ ) s ∏ n = s + 1 N − 1 λ ( 1 + γ ) s μ + ( n − s ) r + N s ! ( λ ( 1 + γ ) μ ) s ∗ ∏ n = s + 1 N λ ( 1 + γ ) s μ + ( n − s ) r ] (8)

6) Expected number of customers waiting in the queue is

L q = ∑ n = s N ( n − s ) P n = P 0 [ ∑ n = s + 1 N − 1 ( n − s ) ( λ ( 1 + γ ) μ ) s ∏ n = s + 1 N λ ( 1 + γ ) s μ + ( n − s ) r + ( N − s ) s ! ( λ ( 1 + γ ) μ ) s ∗ ∏ n = s + 1 N λ ( 1 + γ ) s μ + ( n − s ) r ] (9)

7) Expected waiting time of customers in the system and queue are

W s = L s λ , W q = L q λ (10)

Arrivals of customers, their service and percentage of change of customers in the queueing system remain always uncertain due to the fact that within the set of disjoint time-intervals, the customers may arrive in very slow rate, slow rate, fast arrival rate and very fast arrival rate. The same trait may exist in service rates and in rate of change in customers in the system. Incorporation of such uncertainty characteristics in the queueing model yields more realistic which is possible by the inclusion of the concepts of fuzzy set theory. Fuzzy set theory came into existence only after the fuzzy logic was introduced first by Zadeh [

Let arrival rate λ ˜ = ( λ 1 , λ 2 , λ 3 , λ 4 ) , service rate μ ˜ = ( μ 1 , μ 2 , μ 3 , μ 4 ) , percentage change in number of customers γ ˜ = ( γ 1 , γ 2 , γ 3 , γ 4 ) , reneging rate r ˜ = ( r 1 , r 2 , r 3 , r 4 ) corresponding to attributes very slow, slow, fast, very fast in their respective order of i, 1 ≤ i ≤ 4 such that

λ 1 ≤ λ 2 ≤ λ 3 ≤ λ 4 , μ 1 ≤ μ 2 ≤ μ 3 ≤ μ 4 , γ 1 ≤ γ 2 ≤ γ 3 ≤ γ 4 , r 1 ≤ r 2 ≤ r 3 ≤ r 4

We define membership of T λ ˜ ( λ ) , 1 − F λ ˜ ( λ ) , T μ ˜ ( μ ) , 1 − F μ ˜ ( μ ) , T γ ˜ ( γ ) , 1 − F γ ˜ ( γ ) , T r ˜ ( r ) , 1 − F r ˜ ( r ) as follows:

T λ ˜ ( λ ) = { λ − λ 1 w ( λ 2 − λ 1 ) , λ 1 ≤ λ ≤ λ 2 1 , λ 2 ≤ λ ≤ λ 3 λ 4 − λ w ( λ 4 − λ 3 ) , λ 3 ≤ λ ≤ λ 4 0 , otherwise (11)

1 − F λ ˜ ( λ ) = { λ − λ 1 λ 2 − λ 1 , λ 1 ≤ λ ≤ λ 2 1 , λ 2 ≤ λ ≤ λ 3 λ 4 − λ λ 4 − λ 3 , λ 3 ≤ λ ≤ λ 4 0 , otherwise (12)

T μ ˜ ( μ ) = { μ − μ 1 w ( μ 2 − μ 1 ) , μ 1 ≤ μ ≤ μ 2 1 , μ 2 ≤ μ ≤ μ 3 μ 4 − μ w ( μ 4 − μ 3 ) , μ 3 ≤ μ ≤ μ 4 0 , otherwise (13)

1 − F μ ˜ ( μ ) = { μ − μ 1 μ 2 − μ 1 , μ 1 ≤ μ ≤ μ 2 1 , μ 2 ≤ μ ≤ μ 3 μ 4 − μ μ 4 − μ 3 , μ 3 ≤ μ ≤ μ 4 0 , otherwise (14)

T γ ˜ ( γ ) = { γ − γ 1 w ( γ 2 − γ 1 ) , γ 1 ≤ γ ≤ γ 2 1 , γ 2 ≤ γ ≤ γ 3 γ 4 − γ w ( γ 4 − γ 3 ) , γ 3 ≤ γ ≤ γ 4 0 , otherwise (15)

1 − F γ ˜ ( γ ) = { γ − γ 1 γ 2 − γ 1 , γ 1 ≤ γ ≤ γ 2 1 , γ 2 ≤ γ ≤ γ 3 γ 4 − γ γ 4 − γ 3 , γ 3 ≤ γ ≤ γ 4 0 , otherwise (16)

T r ˜ ( r ) = { r − r 1 w ( r 2 − r 1 ) , r 1 ≤ r ≤ r 2 1 , r 2 ≤ r ≤ r 3 r 4 − r w ( r 4 − r 3 ) , r 3 ≤ r ≤ r 4 0 , otherwise (17)

1 − F r ˜ ( r ) = { r − r 1 r 2 − r 1 , r 1 ≤ r ≤ r 2 1 , r 2 ≤ r ≤ r 3 r 4 − r r 4 − r 3 , r 3 ≤ r ≤ r 4 0 , otherwise (18)

where w ∈ [ 1 , ∞ ) .

By the method of α-cut, we have

λ ˜ − λ 1 w ( λ 2 − λ 1 ) = α T so that λ ˜ = w α T ( λ 2 − λ 1 ) + λ 1

λ 4 − λ ˜ w ( λ 4 − λ 3 ) = α T so that λ ˜ = λ 4 − w α T ( λ 4 − λ 3 )

and

λ ˜ α T = [ w α T ( λ 2 − λ 1 ) + λ 1 , λ 4 − w α T ( λ 4 − λ 3 ) ] (19)

λ ˜ − λ 1 λ 2 − λ 1 = α F so that λ ˜ = α F ( λ 2 − λ 1 ) + λ 1

λ 4 − λ ˜ λ 4 − λ 3 = α F so that λ ˜ = λ 4 − α F ( λ 4 − λ 3 )

and

λ ˜ α F = [ α F ( λ 2 − λ 1 ) + λ 1 , λ 4 − α F ( λ 4 − λ 3 ) ] (20)

Similarly

μ ˜ α T = [ w ˜ α T ( μ 2 − μ 1 ) + μ 1 , μ 4 − w α T ( μ 4 − μ 3 ) ] (21)

μ ˜ α F = [ α F ( µ 2 − µ 1 ) + μ 1 , μ 4 − α F ( μ 4 − μ 3 ) ] (22)

γ ˜ α T = [ w α T ( γ 2 − γ 1 ) + γ 1 , γ 4 − w α T ( γ 4 − γ 3 ) ] (23)

γ ˜ α F = [ α F ( γ 2 − γ 1 ) + γ 1 , γ 4 − α F ( γ 4 − γ 3 ) ] (24)

r ˜ α T = [ w α T ( r 2 − r 1 ) + r 1 , r 4 − w α T ( r 4 − r 3 ) ] (26)

r ˜ α F = [ α F ( r 2 − r 1 ) + r 1 , r 4 − α F ( r 4 − r 3 ) ] (26)

By the definition of vague number, we have

λ ˜ α T = [ λ ˜ α T L , λ ˜ α T U ] , λ ˜ α F = [ λ ˜ α F L , λ ˜ α F U ]

μ ˜ α T = [ μ ˜ α T L , μ ˜ α T U ] , μ ˜ α F = [ μ ˜ α F L , μ ˜ α F U ]

γ ˜ α T = [ γ ˜ α T L , γ ˜ α T U ] , γ ˜ α F = [ γ ˜ α F L , γ ˜ α F U ]

r ˜ α T = [ r ˜ α T L , r ˜ α T U ] , r ˜ α F = [ r ˜ α F L , r ˜ α F U ] (27)

Fuzzy probability of empty system

( P ˜ 0 ) α T L = [ ∑ n = 0 s 1 n ! ( ( w α T ( λ 2 − λ 1 ) + λ 1 ) ( 1 + w α T ( γ 2 − γ 1 ) + γ 1 ) w α T ( μ 2 − μ 1 ) + μ 1 ) n + ∑ n = s + 1 N { 1 s ! ( ( w α T ( λ 2 − λ 1 ) + λ 1 ) ( 1 + w α T ( γ 2 − γ 1 ) + γ 1 ) w α T ( μ 2 − μ 1 ) + μ 1 ) s ∗ ∏ i = s + 1 n ( ( w α T ( λ 2 − λ 1 ) + λ 1 ) ( 1 + w α T ( γ 2 − γ 1 ) + γ 1 ) s ( w α T ( μ 2 − μ 1 ) + μ 1 ) + ( i − s ) ( w α T ( r 2 − r 1 ) + r 1 ) ) } ] − 1 (28)

( P ˜ 0 ) α T U = [ ∑ n = 0 s 1 n ! ( ( λ 4 − w α T ( λ 4 − λ 3 ) ) ( 1 + ( γ 4 − w α T ( γ 4 − γ 3 ) ) μ 4 − w α T ( μ 4 − μ 3 ) ) n + ∑ n = s + 1 N { 1 s ! ( ( λ 4 − w α T ( λ 4 − λ 3 ) ) ( 1 + ( γ 4 − w α T ( γ 4 − γ 3 ) ) μ 4 − w α T ( μ 4 − μ 3 ) ) s ∗ ∏ i = s + 1 n ( ( λ 4 − w α T ( λ 4 − λ 3 ) ) ( 1 + ( γ 4 − w α T ( γ 4 − γ 3 ) ) s ( μ 4 − w α T ( μ 4 − μ 3 ) ) + ( i − s ) ( r 4 − w α T ( r 4 − r 3 ) ) ) } ] − 1 (29)

( P ˜ 0 ) α F L = [ ∑ n = 0 s 1 n ! ( ( α F ( λ 2 − λ 1 ) + λ 1 ) ( 1 + α F ( γ 2 − γ 1 ) + γ 1 ) α F ( μ 2 − μ 1 ) + μ 1 ) n + ∑ n = s + 1 N { 1 s ! ( ( α F ( λ 2 − λ 1 ) + λ 1 ) ( 1 + α F ( γ 2 − γ 1 ) + γ 1 ) α F ( μ 2 − μ 1 ) + μ 1 ) s ∗ ∏ i = s + 1 n ( ( α F ( λ 2 − λ 1 ) + λ 1 ) ( 1 + α F ( γ 2 − γ 1 ) + γ 1 ) s ( α F ( μ 2 − μ 1 ) + μ 1 ) + ( i − s ) ( α F ( r 2 − r 1 ) + r 1 ) ) } ] − 1 (30)

( P ˜ 0 ) α F U = [ ∑ n = 0 s 1 n ! ( ( λ 4 − α F ( λ 4 − λ 3 ) ) ( 1 + ( γ 4 − α F ( γ 4 − γ 3 ) ) μ 4 − α F ( μ 4 − μ 3 ) ) n + ∑ n = s + 1 N { 1 s ! ( ( λ 4 − α F ( λ 4 − λ 3 ) ) ( 1 + ( γ 4 − α F ( γ 4 − γ 3 ) ) μ 4 − α F ( μ 4 − μ 3 ) ) s ∗ ∏ i = s + 1 n ( ( λ 4 − α F ( λ 4 − λ 3 ) ) ( 1 + ( γ 4 − α F ( γ 4 − γ 3 ) ) s ( μ 4 − α F ( μ 4 − μ 3 ) ) + ( i − s ) ( r 4 − α F ( r 4 − r 3 ) ) ) } ] − 1 (31)

Fuzzy probability that system is full is given by

( P ˜ N ) α T L = 1 s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∏ i = s + 1 N λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L ( P ˜ 0 ) α T L (32)

( P ˜ N ) α T U = 1 s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∏ i = s + 1 N λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U ( P ˜ 0 ) α T U (33)

( P ˜ N ) α F L = 1 s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∏ i = s + 1 N λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L ( P ˜ 0 ) α F L (34)

( P ˜ N ) α F U = 1 s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) s ∏ i = s + 1 N λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U ( P ˜ 0 ) α F U (35)

Fuzzy expected system size

( L ˜ s ) α T L = ( P ˜ 0 ) α T L [ ∑ n = 1 s 1 ( n − 1 ) ! ( λ α T L ( 1 + γ α T L ) μ α T L ) n + ∑ n = s + 1 N − 1 n s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∗ ∏ i = s + 1 N − 1 λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L + N s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∏ n = s + 1 N λ α T L ( 1 + γ α T L ) s μ α T L + ( n − s ) r α T L ] (36)

( L ˜ s ) α T U = ( P ˜ 0 ) α T U [ ∑ n = 1 s 1 ( n − 1 ) ! ( λ α T U ( 1 + γ α T U ) μ α T U ) n + ∑ n = s + 1 N − 1 n s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∗ ∏ i = s + 1 N − 1 λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U + N s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∏ n = s + 1 N λ α T U ( 1 + γ α T U ) s μ α T U + ( n − s ) r α T U ] (37)

( L ˜ s ) α F L = ( P ˜ 0 ) α F L [ ∑ n = 1 s 1 ( n − 1 ) ! ( λ α F L ( 1 + γ α F L ) μ α F L ) n + ∑ n = s + 1 N − 1 n s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∗ ∏ i = s + 1 N − 1 λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L + N s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∏ n = s + 1 N λ α F L ( 1 + γ α F L ) s μ α F L + ( n − s ) r α F L ] (38)

( L ˜ s ) α F U = ( P ˜ 0 ) α F U [ ∑ n = 1 s 1 ( n − 1 ) ! ( λ α F U ( 1 + γ α F U ) μ α F U ) n + ∑ n = s + 1 N − 1 n s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) s ∗ ∏ i = s + 1 N − 1 λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U + N s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) s ∏ n = s + 1 N λ α F U ( 1 + γ α F U ) s μ α F U + ( n − s ) r α F U ] (39)

Fuzzy expected number of customers waiting in queue is:

( L ˜ q ) α T L = ( P ˜ 0 ) α T L [ ∑ n = s + 1 N − 1 ( n − s ) ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∏ i = s + 1 N λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L + ( N − s ) s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) ∗ ∏ i = s + 1 N λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L ] (40)

( L ˜ q ) α T U = ( P ˜ 0 ) α T U [ ∑ n = s + 1 N − 1 ( n − s ) ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∏ i = s + 1 N λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U + ( N − s ) s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) ∗ ∏ i = s + 1 N λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U ] (41)

( L ˜ q ) α F L = ( P ˜ 0 ) α F L [ ∑ n = s + 1 N − 1 ( n − s ) ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∏ i = s + 1 N λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L + ( N − s ) s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) ∗ ∏ i = s + 1 N λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L ] (42)

( L ˜ q ) α F U = ( P ˜ 0 ) α F U [ ∑ n = s + 1 N − 1 ( n − s ) ( λ α F U ( 1 + γ α F U ) μ α F U ) s ∏ i = s + 1 N λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U + ( N − s ) s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) ∗ ∏ i = s + 1 N λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U ] (43)

Fuzzy expected time spent in the system is:

( W ˜ s ) α T L = ( L ˜ s ) α T L λ ˜ α T L , ( W ˜ s ) α T U = ( L ˜ s ) α T U λ ˜ α T U (44)

( W ˜ s ) α F L = ( L ˜ s ) α F L λ ˜ α F L , ( W ˜ s ) α F U = ( L ˜ s ) α F U λ ˜ α F U (45)

Fuzzy expected time waiting in queue is:

( W ˜ q ) α T L = ( L ˜ q ) α T L λ ˜ α T L , ( W ˜ q ) α T U = ( L ˜ q ) α T U λ ˜ α T U (46)

( W ˜ q ) α F L = ( L ˜ q ) α F L λ ˜ α F L , ( W ˜ q ) α F U = ( L ˜ q ) α F U λ ˜ α F U (47)

Fuzzy number of idle servers per unit time is:

( E ( I ) ˜ ) α T L = ∑ n = 0 s − 1 ( s − n ) n ! ( λ α T L ( 1 + γ α T L ) μ α T L ) n ( P ˜ 0 ) α T L (48)

( E ( I ) ˜ ) α T U = ∑ n = 0 s − 1 ( s − n ) n ! ( λ α T U ( 1 + γ α T U ) μ α T U ) n ( P ˜ 0 ) α T U (49)

( E ( I ) ˜ ) α F L = ∑ n = 0 s − 1 ( s − n ) n ! ( λ α F L ( 1 + γ α F L ) μ α F L ) n ( P ˜ 0 ) α F L (50)

( E ( I ) ˜ ) α F U = ∑ n = 0 s − 1 ( s − n ) n ! ( λ α F U ( 1 + γ α F U ) μ α F U ) n ( P ˜ 0 ) α F U (51)

Fuzzy expected number of busy servers per unit time is:

( E ( B ) ˜ ) α T L = s − ( E ( I ) ˜ ) α T L (52)

( E ( B ) ˜ ) α T U = s − ( E ( I ) ˜ ) α T U (53)

( E ( B ) ˜ ) α F L = s − ( E ( I ) ˜ ) α F L (54)

( E ( B ) ˜ ) α F U = s − ( E ( I ) ˜ ) α F U (55)

Probability that server remains busy is

( P ( B ) ˜ ) α T L = 1 s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s + ∑ i = s + 1 N − 1 1 s ! ( λ α T L ( 1 + γ α T L μ α T L ) ) s ∗ ∏ i = s + 1 N ( λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L ) ( P ˜ 0 ) α T L + 1 s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∏ i = s + 1 N ( λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L ) ( P ˜ 0 ) α T L (56)

( P ( B ) ˜ ) α T U = 1 s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s + ∑ i = s + 1 N − 1 1 s ! ( λ α T U ( 1 + γ α T U μ α T U ) ) s ∗ ∏ i = s + 1 N ( λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U ) ( P ˜ 0 ) α T U + 1 s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∏ i = s + 1 N ( λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U ) ( P ˜ 0 ) α T U (57)

( P ( B ) ˜ ) α F L = 1 s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s + ∑ i = s + 1 N − 1 1 s ! ( λ α F L ( 1 + γ α F L μ α F L ) ) s ∗ ∏ i = s + 1 N ( λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L ) ( P ˜ 0 ) α F L + 1 s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∏ i = s + 1 N ( λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L ) ( P ˜ 0 ) α F L (58)

( P ( B ) ˜ ) α F U = 1 s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) s + ∑ i = s + 1 N − 1 1 s ! ( λ α F U ( 1 + γ α F U μ α F U ) ) s ∗ ∏ i = s + 1 N ( λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U ) ( P ˜ 0 ) α F U + 1 s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) s ∏ i = s + 1 N ( λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U ) ( P ˜ 0 ) α F U (59)

Fuzzy reneging rate is

( R ˜ r ) α T L = [ ∑ n = s + 1 N − 1 ( n − s ) s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∏ i = s + 1 N λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L + ( N − s ) s ! ( λ α T L ( 1 + γ α T L ) μ α T L ) s ∏ i = s + 1 N λ α T L ( 1 + γ α T L ) s μ α T L + ( i − s ) r α T L ] ( P ˜ 0 ) α T L (60)

( R ˜ r ) α T U = [ ∑ n = s + 1 N − 1 ( n − s ) s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∏ i = s + 1 N λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U + ( N − s ) s ! ( λ α T U ( 1 + γ α T U ) μ α T U ) s ∏ i = s + 1 N λ α T U ( 1 + γ α T U ) s μ α T U + ( i − s ) r α T U ] ( P ˜ 0 ) α T U (61)

( R ˜ r ) α F L = [ ∑ n = s + 1 N − 1 ( n − s ) s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∏ i = s + 1 N λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L + ( N − s ) s ! ( λ α F L ( 1 + γ α F L ) μ α F L ) s ∏ i = s + 1 N λ α F L ( 1 + γ α F L ) s μ α F L + ( i − s ) r α F L ] ( P ˜ 0 ) α F L (62)

( R ˜ r ) α F U = [ ∑ n = s + 1 N − 1 ( n − s ) s ! ( λ α F U ( 1 + γ α U U ) μ α U U ) s ∏ i = s + 1 N λ α U U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U + ( N − s ) s ! ( λ α F U ( 1 + γ α F U ) μ α F U ) s ∏ i = s + 1 N λ α F U ( 1 + γ α F U ) s μ α F U + ( i − s ) r α F U ] ( P ˜ 0 ) α F U (63)

The queueing model studied under fuzzy environment yields various fuzzy parameters which are uncertain in nature. Such an uncertainty be resolved to some extent by using fuzzy optimization technique for which the fuzzy objective function has been constructed. The strategy of minimization of the total cost of the operating horizon is termed as the optimal policy. Taking parameter vector of s, N, λ, μ, L_{q} as decision variables, we develop the steady-state expected total cost function per unit time for M/M/s/N queueing system then convert the optimal policy into fuzzy environment. Our main goal is to find optimal number of servers s* for which following cost parameters are defined in vector form as:

C_{q} = cost per unit time when one customer is waiting for service.

C_{s} = cost per unit time when one customer joins the system and is served according to first-come-first-served discipline.

(C_{I}, C_{B}) = cost per unit time when one server is (idle, busy).

C_{R} = cost of reneging per customer per unit time in queue.

Minimize TEC function

F ( s , N ) = C q L q + C s ( L s − L q ) + C B E ( B ) + C I E ( I ) + C R R r = C q ∗ L q + C s ∗ μ + C B ∗ E ( B ) + C I ∗ E ( I ) + C R ∗ R r (64)

Fuzzy total expected cost functions of the system are:

( T E C ˜ ) α T L = C q ∗ ( L ˜ q ) α T L + C s ∗ μ α T L + C B ∗ ( E ( B ) ˜ ) α T L + C I ∗ ( E ( I ) ˜ ) α T L + C R ∗ ( R ˜ r ) α T L (65)

( T E C ˜ ) α T U = C q ∗ ( L ˜ q ) α T U + C s ∗ μ α T U + C B ∗ ( E ( B ) ˜ ) α T U + C I ∗ ( E ( I ) ˜ ) α T U + C R ∗ ( R ˜ r ) α T U (66)

( T E C ˜ ) α F L = C q ∗ ( L ˜ q ) α F L + C s ∗ μ α F L + C B ∗ ( E ( B ) ˜ ) α F L + C I ∗ ( E ( I ) ˜ ) α F L + C R ∗ ( R ˜ r ) α F L (67)

( T E C ˜ ) α F U = C q ∗ ( L ˜ q ) α F U + C s ∗ μ α F U + C B ∗ ( E ( B ) ˜ ) α F U + C I ∗ ( E ( I ) ˜ ) α F U + C R ∗ ( R ˜ r ) α F U (68)

Above expected cost functions are non-linear due to upper limit of summation sign used in the expressions with the help of which optimal number of servers s* has to be determined.

Let us consider the car-workshop with capacity of N = 8 cars at a time which are repaired/inspected by multiple mechanics and we find various following performance measures of the system. Cars rush to the system to get repaired or to get inspection by the work-shop mechanics in the order of their arrival of first-come-first-served basis. We also find the optimal number of mechanics that can be employed so as to get minimum system costs under the smart service in the sense that cars have minimum time to wait and minimum time spent in the work-shop. _{T} increases from 0.10 to 0.50 the number of customers

S | ( L s ) α T L | ( L s ) α T U | ( L q ) α T L | ( L q ) α T U | ( W s ) α T L | ( W s ) α T U | ( W q ) α T L | ( W q ) α T U | ( E ( I ) ) α T L | ( E ( I ) ) α T U | ( E ( B ) ) α T L | ( E ( B ) ) α T U | ( R r ) α T L | ( R r ) α T U | ( T E C ) α T L | ( T E C ) α T U |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 0.0773 | 0.6493 | 0.5404 | 1.1301 | 0.1610 | 1.0472 | 1.1259 | 1.8228 | 0.0064 | 0.0051 | 1.9936 | 1.9949 | 1338.2 | 1117.8 | 402.7329 | 336.7756 |

3 | 0.0862 | 0.2602 | 0.7826 | 2.2677 | 0.1795 | 0.4197 | 1.6304 | 3.6576 | 0.0223 | 0.0201 | 2.9777 | 2.9799 | 2.6043 | 4.5183 | 2.5898 | 3.5170 |

4 | 0.1376 | 0.1958 | 1.1794 | 4.2827 | 0.2867 | 0.3159 | 2.4571 | 6.9076 | 0.0668 | 0.0649 | 3.9332 | 3.9351 | 0.0682 | 0.1624 | 2.4083 | 3.1133 |

5 | 0.2593 | 0.2814 | 1.4900 | 6.5412 | 0.5402 | 0.4539 | 3.1041 | 10.5503 | 0.1796 | 0.1809 | 4.8204 | 4.8191 | 0.0088 | 0.0224 | 2.9526 | 4.0230 |

6 | 0.4973 | 0.5097 | 1.1287 | 6.2090 | 1.0360 | 0.8220 | 2.3514 | 10.0145 | 0.4404 | 0.4484 | 5.5596 | 5.5516 | 0.0016 | 0.0042 | 3.3782 | 4.4511 |

7 | 0.8991 | 0.8944 | 2.0469e^{−4} | 3.9831e^{−}^{5} | 1.8731 | 0.8220 | 4.2643e^{−4} | 6.4243e^{−}^{5} | 0.9698 | 0.9717 | 6.0302 | 6.0283 | 5.8373e^{−4} | 0.0012 | 3.6522 | 3.7084 |

S | ( L s ) α T L | ( L s ) α T U | ( L q ) α T L | ( L q ) α T U | ( W s ) α T L | ( W s ) α T U | ( W q ) α T L | ( W q ) α T U | ( E ( I ) ) α T L | ( E ( I ) ) α T U | ( E ( B ) ) α T L | ( E ( B ) ) α T U | ( R r ) α T L | ( R r ) α T U | ( T E C ) α T L | ( T E C ) α T U |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 2.4234 | 0.0728 | 1.0238 | 0.7581 | 3.0293 | 0.2428 | 1.2797 | 2.5271 | 0.0071 | 0.0106 | 1.9929 | 1.9886 | 4868.6 | 1126.0 | 1462.1 | 339.0211 |

3 | 0.6506 | 0.0843 | 2.0321 | 1.4102 | 0.8133 | 0.2809 | 2.5402 | 4.7006 | 0.0300 | 0.0332 | 2.9700 | 2.9668 | 15.5909 | 2.1583 | 6.8637 | 2.5095 |

4 | 0.2984 | 0.1376 | 3.8139 | 2.5868 | 0.3730 | 0.4586 | 4.7673 | 8.6228 | 0.1013 | 0.0941 | 3.8987 | 3.9059 | 0.2884 | 0.0434 | 3.1293 | 2.6104 |

5 | 0.3134 | 0.2617 | 5.6171 | 4.0520 | 0.3917 | 0.8722 | 7.0214 | 13.5065 | 0.2894 | 0.2455 | 4.7106 | 4.7545 | 0.0349 | 0.0052 | 3.9139 | 3.3920 |

6 | 0.5221 | 0.5015 | 4.7386 | 4.2185 | 0.6527 | 1.6718 | 5.9195 | 14.0617 | 0.7154 | 0.5899 | 5.2846 | 5.4101 | 0.0068 | 9.6732e^{−4} | 4.2292 | 3.9240 |

7 | 0.8593 | 0.8908 | 3.3222e^{−4} | 0.2293 | 1.0741 | 2.9693 | 4.1527e^{−4} | 0.7643 | 1.4768 | 1.2635 | 5.5232 | 5.7365 | 0.0020 | 0.0005 | 3.7807 | 3.6260 |

in the system and in the line decrease whereas the expected time spent in the system and the waiting time in queue increase. Moreover, expected time spent in the system and the waiting time in queue decrease with the increase of number of servers which is up to our expectation. The expected number of idle servers and expected number of busy servers are increasing with the increase of number of servers and value of α_{T}. Rate of reneging is decreasing with the increase of number of servers which seems quite natural. As α_{T} increases from 0.10 to 0.50 the expected cost of the system increase but it decreases with increase of the number of servers. _{F} increases from 0.12 to 0.50 number of idle servers, number of busy servers, reneging rate of customers and system expected cost decreases significantly. _{T} from 0.10 to 0.50. The same are increasing when α_{F} from 0.12 to 0.50 which has been illustrated in _{T} and α_{F} have been used.

For λ = 0.5 , λ 1 = 0.4 , λ 2 = 0.5 , λ 3 = 0.6 , λ 4 = 0.7 , γ = 0.5 , W = 8 , μ 1 = 0.3 , μ 2 = 0.4 , μ 3 = 0.5 , μ 4 = 0.6 , γ 1 = 0.3 , γ 2 = 0.4 , γ 3 = 0.5 , γ 4 = 0.6 , r 1 = 0.1 , r 2 = 0.2 , r 3 = 0.4 , r 4 = 0.5 , N = 8 .

S | ( L s ) α F L | ( L s ) α F U | ( L q ) α F L | ( L q ) α F U | ( W s ) α F L | ( W s ) α F U | ( W q ) α F L | ( W q ) α F U | ( E ( I ) ) α F L | ( E ( I ) ) α F U | ( E ( B ) ) α F L | ( E ( B ) ) α F U | ( R r ) α F L | ( R r ) α F U | ( T E C ) α F L | ( T E C ) α F U |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 0.1258 | 56.0258 | 0.8302 | 1.1139 | 0.2537 | 92.7580 | 1.6738 | 1.8443 | 0.0114 | 0.0087 | 1.9886 | 1.9913 | 1,142,600 | 9,117,100 | 342,770 | 2,735,100 |

3 | 0.0748 | 26.9658 | 1.5222 | 2.2127 | 0.1508 | 44.6453 | 3.0689 | 3.6635 | 0.0429 | 0.0339 | 2.9571 | 2.9661 | 7829.7 | 192950 | 2350.9 | 57888 |

4 | 0.0410 | 21.4032 | 2.7405 | 4.1460 | 0.0828 | 35.4358 | 5.5252 | 6.8643 | 0.1348 | 0.1086 | 3.8652 | 3.8914 | 68.3530 | 3039.2 | 23.2124 | 914.7909 |

5 | 0.0213 | 15.0145 | 4.1574 | 6.3018 | 0.0429 | 24.8584 | 8.3818 | 10.4335 | 0.3738 | 0.3020 | 4.6262 | 4.6980 | 1.3208 | 57.4999 | 3.8861 | 21.2119 |

6 | 0.0122 | 7.0656 | 4.0766 | 5.9730 | 0.0246 | 11.6980 | 8.2189 | 9.8890 | 0.9345 | 0.7482 | 5.0655 | 5.2518 | 0.0792 | 1.7424 | 3.9975 | 4.9189 |

7 | 0.0031 | 2.1651 | 2.0736e^{−4} | 2.3283e^{−4} | 0.0063 | 3.5845 | 4.1806e^{−4} | 3.8548e^{−4} | 2.0809 | 1.6261 | 4.9191 | 5.3739 | 0.0252 | 0.1512 | 3.6660 | 3.7470 |

S | ( L s ) α F L | ( L s ) α F U | ( L q ) α F L | ( L q ) α F U | ( W s ) α F L | ( W s ) α F U | ( W q ) α F L | ( W q ) α F U | ( E ( I ) ) α F L | ( E ( I ) ) α F U | ( E ( B ) ) α F L | ( E ( B ) ) α F U | ( R r ) α F L | ( R r ) α F U | ( T E C ) α F L | ( T E C ) α F U |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 4.8197 | 282.4809 | 1.2282 | 0.6700 | 6.0246 | 941.6031 | 1.5352 | 2.2332 | 0.0039 | 0.0063 | 1.9961 | 1.9937 | 88057 | 1.6689 | 26419 | 1.7147 |

3 | 1.2122 | 68.5848 | 2.6843 | 1.2164 | 1.5153 | 228.6158 | 3.3553 | 4.0447 | 0.0154 | 0.0199 | 2.9846 | 2.9801 | 406.6311 | 0.1742 | 124.3062 | 1.8755 |

4 | 0.3657 | 35.9664 | 5.4781 | 2.2588 | 0.4572 | 119.8880 | 6.8477 | 7.5293 | 0.0497 | 0.0565 | 3.9503 | 3.9435 | 2.4433 | 0.0278 | 4.1083 | 2.5401 |

5 | 0.1142 | 19.6213 | 8.9237 | 3.6263 | 0.1427 | 65.4044 | 11.1546 | 12.0875 | 0.1380 | 0.1473 | 4.8620 | 4.8527 | 0.0720 | 0.0048 | 4.5863 | 3.3067 |

6 | 0.0351 | 8.0860 | 8.8184 | 3.8155 | 0.0439 | 26.9532 | 11.0230 | 12.7182 | 0.3348 | 0.3539 | 5.6652 | 5.6461 | 0.0096 | 9.2670e^{−4} | 5.0466 | 3.8434 |

7 | 0.0100 | 2.4849 | 3.3222e^{−4} | 2.3910e^{−4} | 0.0125 | 8.2830 | 4.1527e^{−4} | 7.9700e^{−4} | 0.6822 | 0.7581 | 6.3178 | 6.2419 | 0.0043 | 2.2004e^{−4} | 3.7814 | 3.5801 |

s | ( P 0 ) α T L | ( P 0 ) α T U | ( P N ) α T L | ( P N ) α T U | ( P ( B ) ) α T L | ( P ( B ) ) α T U |
---|---|---|---|---|---|---|

2 | 0.0039 | 0.0030 | 6.2227e^{−}^{16} | 1.0271e^{−}^{17} | 0.1604 | 0.1126 |

3 | 0.0069 | 0.0058 | 8.5766e^{−}^{14} | 7.3221e^{−}^{14} | 0.1231 | 0.0945 |

4 | 0.0133 | 0.0118 | 8.8824e^{−}^{12} | 1.6875e^{−}^{10} | 0.0744 | 0.0596 |

5 | 0.0260 | 0.0236 | 1.0714e^{−}^{9} | 8.9744e^{−}^{8} | 0.0342 | 0.0274 |

6 | 0.0500 | 0.0456 | 2.5599e^{−}^{7} | 1.0785e^{−}^{5} | 0.0114 | 0.0088 |

7 | 0.0904 | 0.0808 | 2.0469e^{−}^{4} | 3.495e^{−}^{4} | 0.0023 | 0.0203 |

s | ( P 0 ) α T L | ( P 0 ) α T U | ( P N ) α T L | ( P N ) α T U | ( P ( B ) ) α T L | ( P ( B ) ) α T U |
---|---|---|---|---|---|---|

2 | 0.0024 | 0.0041 | 7.1980e^{−}^{17} | 6.2017e^{−}^{19} | 0.2454 | 0.0657 |

3 | 0.0048 | 0.0068 | 5.7800e^{−}^{14} | 1.0685e^{−}^{15} | 0.2247 | 0.0503 |

4 | 0.0099 | 0.0126 | 1.5664e^{−}^{11} | 2.1512e^{−}^{12} | 0.1536 | 0.0314 |

5 | 0.0200 | 0.0243 | 2.7290e^{−}^{9} | 2.0860e^{−}^{9} | 0.0756 | 0.0152 |

6 | 0.0380 | 0.0463 | 6.0886e^{−}^{7} | 6.9261e^{−}^{7} | 0.0252 | 0.0055 |

7 | 0.0636 | 0.0820 | 3.3222e^{−}^{4} | 7.1730e^{−}^{5} | 0.0046 | 0.0208 |

s | ( P 0 ) α F L | ( P 0 ) α F U | ( P N ) α F L | ( P N ) α F U | ( P ( B ) ) α F L | ( P ( B ) ) α F U |
---|---|---|---|---|---|---|

2 | 0.0039 | 0.0031 | 7.2572e^{−}^{18} | 5.2839e^{−}^{18} | 0.0819 | 0.1107 |

3 | 0.0068 | 0.0059 | 1.5420e^{−}^{14} | 3.3246e^{−}^{14} | 0.0632 | 0.0920 |

4 | 0.0132 | 0.0120 | 2.3120e^{−}^{11} | 7.6837e^{−}^{11} | 0.0381 | 0.0577 |

5 | 0.0258 | 0.0240 | 1.3186e^{−}^{18} | 4.4998e^{−}^{8} | 0.0174 | 0.0264 |

6 | 0.0496 | 0.0462 | 2.3190e^{−}^{6} | 6.3901e^{−}^{6} | 0.0059 | 0.0085 |

7 | 0.0896 | 0.0822 | 1.2077e^{−}^{4} | 2.3283e^{−}^{4} | 0.0220 | 0.0205 |

s | ( P 0 ) α F L | ( P 0 ) α F U | ( P N ) α F L | ( P N ) α F U | ( P ( B ) ) α F L | ( P ( B ) ) α F U |
---|---|---|---|---|---|---|

2 | 0.0024 | 0.0041 | 1.3678e^{−}^{17} | 1.1447e^{−}^{16} | 0.1221 | 0.0657 |

3 | 0.0048 | 0.0068 | 1.3382e^{−}^{13} | 1.5213e^{−}^{13} | 0.1116 | 0.0503 |

4 | 0.0099 | 0.0126 | 3.5805e^{−}^{10} | 1.5628e^{−}^{10} | 0.0762 | 0.0314 |

5 | 0.0200 | 0.0243 | 1.9504e^{−}^{7} | 6.1529e^{−}^{8} | 0.0374 | 0.0152 |

6 | 0.0380 | 0.0463 | 2.1847e^{−}^{5} | 7.2001e^{−}^{6} | 0.0125 | 0.0055 |

7 | 0.0636 | 0.0820 | 5.3910e^{−}^{4} | 2.3910e^{−}^{4} | 0.0182 | 0.0208 |

α T | S ∗ α T L | ( W S ) α T L | ( W q ) α T L | ( T E C ) α T L |
---|---|---|---|---|

0.10 | 2 | 0.1610 | ||

2 | 1.1259 | |||

2 | 2.4083 | |||

0.50 | 4 | 0.3730 | ||

3 | 2.5402 | |||

4 | 3.1293 |

α T | S ∗ α T U | ( W S ) α T U | ( W q ) α T U | ( T E C ) α T U |
---|---|---|---|---|

0.10 | 4 | 0.3159 | ||

4 | 6.4243e^{−}^{5} | |||

4 | 3.1133 | |||

0.50 | 2 | 0.2428 | ||

7 | 2.7643 | |||

3 | 2.5095 |

α F | S ∗ α F L | ( W S ) α F L | ( W q ) α F L | ( T E C ) α F L |
---|---|---|---|---|

0.50 | 7 | 0.0125 | ||

2 | 1.5352 | |||

7 | 3.7814 | |||

0.12 | 7 | 0.0063 | ||

7 | 4.1806e^{−}^{4} | |||

7 | 3.6660 |

α F | S ∗ α F U | ( W S ) α F U | ( W q ) α F U | ( T E C ) α F U |
---|---|---|---|---|

0.50 | 7 | 8.2830 | ||

7 | 7.9700e^{−}^{4} | |||

2 | 1.7147 | |||

0.12 | 7 | 3.5845 | ||

7 | 3.8548e^{−}^{4} | |||

7 | 3.7470 |

We have developed the queueing model under fuzzy environment and made its intensive studies. The various performance measures obtained have been tabular forms. Also, we have made optimization of fuzzy queueing system for its optimal number of servers with respect to minimization of system costs which reveals that minimization of costs exists generally at threshold values of server. The model studied under fuzzy environment may have widespread applications in artificial intelligence, machine design, and robot and robotic and in many business promotion prospects.

The authors declare no conflicts of interest regarding the publication of this paper.

Panta, A.P., Ghimire, R.P., Panthi, D. and Pant, S.R. (2021) Optimization of M/M/s/N Queueing Model with Reneging in a Fuzzy Environment. American Journal of Operations Research, 11, 121-140. https://doi.org/10.4236/ajor.2021.113008