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r queues with FIFO
Volume 2, Number 1
OPERATIONSRESEARCHLETTERS
DECOMPOSITION OF AN M/D/r,k QUEUES WITH FIFO *
April 1983
QUEUE WITH FIFO INTO k Ek/D/r
ViUy Baek IVERSEN The Institute of Mathematical Statistics and Operations Research, The Technical University of Denmark, DK.2800 Lyngby, Denmark ReceivedOctober 1982 Revised February 1983
it is observed that the queuing system M/D/r. k with FIFO has the same waitingtime distribution as the queuing system Ek/D/r with FIFO. Usingthis simpleequivalencewe can apply numericalmethods and tables for M/D/n to solve Et/D/r. E~/D/r queue with FIFO
Introduction In the literature, the queuing system with constant holding times, G I l D / n , has usually been dealt with as a special case of the more general system G I / G / n by letting the general holding time distribution (G) equal to a constant (D). This approach is useful in many cases, but systems with constant holding times have some distinct features which are ignored by this general approach.
I, G! / D / n with FIFO The basic property of queuing systems with constant holding times is that the customers depart from the group of servers in the same order as they are accepted for service. If we furthermore assume (a) that the queuing discipline is FIFO and (b) that during non-busy periods the servers also are assigned in a cyclic way I, 2 , . . . , n -
l , n , !, 2,...
If n is an integral multiple of k (i.e. n - k . r, k and r integers), then a group of serverg composed of servers numbers x,x+k,x+2.k,
.... x + ( r - I ) k
(0
(1) will serve exactly every k th customer. That is, if we consider the servers (I), then this group of r servers is equivalent to the queuing system G l k * / D / r , where the interarrival time distribution GI k" is a k-fold convolution of the interarrival time distribution GI. The same holds for the other k - i possible systems. The traffic flows of these k systems are mutually correlated. But if we only consider one system at a time, then this will be a G! k * / D / r queue with FIFO. Thus the waiting time distributions of these two systems are identical. The assumption (b) that the servers are assigned in a cyclic way is not necessary within the group of the r servers. The assumption (a) that the queue discipline is FIFO is only necessary when we consider the waiting time distribution. State probabilities, mean waiting times, etc. are independent of the queue discipline.
(n = number of servers), then a specific server will serve exactly every n th customer. No customer can overtake any other customer.
2. M / D / r with FIFO
* This research has been supported by the Danish Technical Science Research Council, Grant 16-0197.
If we let the interarrival time distribution (GI) equal to the exponential distribution (M), then
20
0167-6377/83/$03.00 © 1983 ElsevierScience Publishers B.V.(North-Holland)
Volume ?, Number I
OPERATIONS RESEARCH LETTERS
GI k• is equal to the Erlang-k distribution (Ek). Thus we get that the queuing system M/D/r. k with FIFO is equivalent ~o the queuing system E J D / r with FIFO, and we can use a table for the M/D/n system to obtain the waiting time distribution of the EJD/r system (n ffi k. r). This was noticed by Pollaczek [2] for r - 1 , but by the above-mentioned interpretation we are able to deal with the system for any r in a very simple way. The Ek/D/r queue has only recently been dealt with by Xerocostas and Demertzes [3]; however, the waiting time distribution for the FIFO queue discipline was not derived by them, and they did not notice the equivalence with the M/D/n queue. In general we know that the performance of a M/D/n system for a given traffic per server ( < 1) improves as n increases (economy of scale). For the same reason, the performance of a queuing system must improve as the arrival process becomes more regular. This can be observed by the above decomposition, where the arrival process of E j D / r becomes more regular as k increases (r constant). Numerical calculations for a M/D/n system are relatively simple, especially as it concerns the steady-state probabilities, which yields the waiting time distribution for integral waiting times (Iversen [!]). Thus we have for M/D/n the waiting time distribution
April 1983
where t is any non-negativeinteger, and P(i) is the steady-state probability of / customers in the system. We have shown that M/D/r. k has the same waiting time distribution as Ek/D/r , and therefore for this system we get rk(t+ l ) - i
P~{W<~I):
E
P(i).
(3)
i..O
By the above decomposition we can immediately apply exact solutions, approximations, and tables for M/D/n to E J D / r and vice versa. References [1] V.B. Iversen, "Exact calculation of waiting time distribu. tions in queuing systems with constant holding times", Fourth Nordic Teletrafflc Seminar ( NT$ IV), Helsinki, May 1!-13 (1982). [2] F. Pollaczek, "Application de ia thi~orie des pgobabilit6s a des probl#mes pos#s par rencombrement des r#seaux t~.16phoniques", Ann. T~l~communications 14, 165-183
(1959). [3] D.A. Xerocostas and C. Demertzes, "Steady state solution of the E s / D / r queuing model", OR.$pektrum 4, ,,7-51 (1982).
n(t+l)-I
E
p(i)
(2)
i-O
21
OPERATIONSRESEARCHLETTERS
DECOMPOSITION OF AN M/D/r,k QUEUES WITH FIFO *
April 1983
QUEUE WITH FIFO INTO k Ek/D/r
ViUy Baek IVERSEN The Institute of Mathematical Statistics and Operations Research, The Technical University of Denmark, DK.2800 Lyngby, Denmark ReceivedOctober 1982 Revised February 1983
it is observed that the queuing system M/D/r. k with FIFO has the same waitingtime distribution as the queuing system Ek/D/r with FIFO. Usingthis simpleequivalencewe can apply numericalmethods and tables for M/D/n to solve Et/D/r. E~/D/r queue with FIFO
Introduction In the literature, the queuing system with constant holding times, G I l D / n , has usually been dealt with as a special case of the more general system G I / G / n by letting the general holding time distribution (G) equal to a constant (D). This approach is useful in many cases, but systems with constant holding times have some distinct features which are ignored by this general approach.
I, G! / D / n with FIFO The basic property of queuing systems with constant holding times is that the customers depart from the group of servers in the same order as they are accepted for service. If we furthermore assume (a) that the queuing discipline is FIFO and (b) that during non-busy periods the servers also are assigned in a cyclic way I, 2 , . . . , n -
l , n , !, 2,...
If n is an integral multiple of k (i.e. n - k . r, k and r integers), then a group of serverg composed of servers numbers x,x+k,x+2.k,
.... x + ( r - I ) k
(0
(1) will serve exactly every k th customer. That is, if we consider the servers (I), then this group of r servers is equivalent to the queuing system G l k * / D / r , where the interarrival time distribution GI k" is a k-fold convolution of the interarrival time distribution GI. The same holds for the other k - i possible systems. The traffic flows of these k systems are mutually correlated. But if we only consider one system at a time, then this will be a G! k * / D / r queue with FIFO. Thus the waiting time distributions of these two systems are identical. The assumption (b) that the servers are assigned in a cyclic way is not necessary within the group of the r servers. The assumption (a) that the queue discipline is FIFO is only necessary when we consider the waiting time distribution. State probabilities, mean waiting times, etc. are independent of the queue discipline.
(n = number of servers), then a specific server will serve exactly every n th customer. No customer can overtake any other customer.
2. M / D / r with FIFO
* This research has been supported by the Danish Technical Science Research Council, Grant 16-0197.
If we let the interarrival time distribution (GI) equal to the exponential distribution (M), then
20
0167-6377/83/$03.00 © 1983 ElsevierScience Publishers B.V.(North-Holland)
Volume ?, Number I
OPERATIONS RESEARCH LETTERS
GI k• is equal to the Erlang-k distribution (Ek). Thus we get that the queuing system M/D/r. k with FIFO is equivalent ~o the queuing system E J D / r with FIFO, and we can use a table for the M/D/n system to obtain the waiting time distribution of the EJD/r system (n ffi k. r). This was noticed by Pollaczek [2] for r - 1 , but by the above-mentioned interpretation we are able to deal with the system for any r in a very simple way. The Ek/D/r queue has only recently been dealt with by Xerocostas and Demertzes [3]; however, the waiting time distribution for the FIFO queue discipline was not derived by them, and they did not notice the equivalence with the M/D/n queue. In general we know that the performance of a M/D/n system for a given traffic per server ( < 1) improves as n increases (economy of scale). For the same reason, the performance of a queuing system must improve as the arrival process becomes more regular. This can be observed by the above decomposition, where the arrival process of E j D / r becomes more regular as k increases (r constant). Numerical calculations for a M/D/n system are relatively simple, especially as it concerns the steady-state probabilities, which yields the waiting time distribution for integral waiting times (Iversen [!]). Thus we have for M/D/n the waiting time distribution
April 1983
where t is any non-negativeinteger, and P(i) is the steady-state probability of / customers in the system. We have shown that M/D/r. k has the same waiting time distribution as Ek/D/r , and therefore for this system we get rk(t+ l ) - i
P~{W<~I):
E
P(i).
(3)
i..O
By the above decomposition we can immediately apply exact solutions, approximations, and tables for M/D/n to E J D / r and vice versa. References [1] V.B. Iversen, "Exact calculation of waiting time distribu. tions in queuing systems with constant holding times", Fourth Nordic Teletrafflc Seminar ( NT$ IV), Helsinki, May 1!-13 (1982). [2] F. Pollaczek, "Application de ia thi~orie des pgobabilit6s a des probl#mes pos#s par rencombrement des r#seaux t~.16phoniques", Ann. T~l~communications 14, 165-183
(1959). [3] D.A. Xerocostas and C. Demertzes, "Steady state solution of the E s / D / r queuing model", OR.$pektrum 4, ,,7-51 (1982).
n(t+l)-I
E
p(i)
(2)
i-O
21